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Full text of "Elementary trigonometry"

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ELEMENTARY 
TRIGONOMETRY 

BY 

W. E. PATERSOX, M.A., B.Sc. 

MATHEMATICAL MASTER, SIERCERs' SCHOOL 
AUTHOR OF ' SCHOOL ALGEBRA 



OXFORD 

AT THE CLARENDON PRESS 

LONDON, EDINBURGH, NEW YORK, TORONTO 

AND MELBOURNE 

HENRY FROWDE 

1911 



BY THE SAME AUTHOR 

SCHOOL ALGEBRA 

THIRD EDITION 



The complete book .... 


Without 
Answers, j 

4s. 


With 

(Answers. 

5s. 


In Parts — 










Part I (Chapters I-XXII : to In- 
dices and Logarithms, including 
Fractions and Quadratic Equa- 
tions) ..... 


2s. 


6d. 


3s. 




Part II (Chapters XXIII-XXXIV: 

Progressions and Ratio to Loga- 
rithmic Series, Theory of Equa- 
tions and Theory of Numbers) . 


2s. 


6d. 


3s. 




In Sections — 

Section A (Chapters I-XVII : to 
Quadratic Equations) . 


2s. 




2s. 


4d, 


Section B (Chapters , XVIII- 
XXVIjLl: to Binomial Theorem 
for Positive Integral Index) , 


2s. 




2s. 


4d. 


Section C (Chapters' XXIX- 
XXXIV: Series, Partial Frac- 
tions, Theory of Equations, De- 
terminants and Theory of Num- 
bers) ... ... 


2s. 


3d. 


2s. 


6d. 


Sections A and B: 'Elementary 
Algebra' ..... 


3s. 


6d. 


4s. 




Examples only— Section A . 


Is. 


3d. 


Is. 


6d. 



PREFACE 

The common examination syllabus ' Trigonometry up 
to and including solution of triangles ' has caused most 
textbooks to treat the subject as if the sole use of 
Trigonometry were to solve triangles, and the practical 
examples deal almost exclusively with various forms of 
triangle-solving under the heading ' Heights and Dis- 
tances '. Further it is customary to define the trigono- 
metrical ratios by means of a right-angled triangle; 
this encourages the mistaken idea that the ratios are 
fundamentally attached to a triangle, and does not 
impress upon the pupil the fact that they are the pro- 
perty of an angle and of an angle only. 

In this book the trigonometrical ratios are introduced 
as functions of the angle. The trigonometrical proper- 
ties of the single angle are treated fully in the early 
chapters, and from the beginning the examples apply 
Trigonometry wherever it may be useful, to Geometry, 
Mensuration, Analytical Geometry, Physical formulae, 
&c. The right-angled triangle definitions are given in 
Chapter Y. This chapter contains, in addition to the 
usual matter, a short treatment of Plane Sailing in 
Navigation. It is hoped that the examples in Navigation 
will provide practice in the use of tables, and at the 
same time be of interest to the pupil. Other examples 

260011 



4 PREFACE 

in this chapter lead up to the formulae dealing with the 
ordinary triangle. 

The formulae for ^ A are proved by Geometry, indepen- 
dently of tlie addition formulae; thus the triangle is 
treated fully without breaking the sequence with a dis- 
cussion of the A + B and allied formulae. 

A chapter is devoted to Projection. This includes 
a discussion of Vector Quantities, their composition and 
resolution, and finishes with a geometrical treatment 
of Demoivre's Theorem. In the last chapter, the addi- 
tion formulae and the allied formulae are treated fully ; 
the projection proofs are used and recommended but the 
old-fashioned proofs are also given 

Throughout the book the student is given every oppor- 
tunity of developing the subject for himself. A large 
portion of the bookwork first occurs among the examples 
of earlier chapters. Also, when a formula has been 
proved, the proofs of others of the same kind are left 
for the student to supply. Thus, when siii^A + cos^^ = 1 
has been proved, the student should have no difficulty 
in proving the connexion between sec^A and tsui^A; 
when sin(^ + ^) has been found, the student should 
himself find the expanded form of cos (J. + B), Sec. 

The sets of examples in the body of a chapter arc num- 
bered IV. a, IV. b, &c. ; these deal only with the matter 
immediately preceding them. The last set of examples in 
a chapter has no distinguishing letter and serves for 
revision of the whole chapter. There are also three 
sets of Revision Examples, l^ookwork is frequentl}^ set 
as an example, both in the Revision Sets and elsewhere ; 
only by constant repetition, oral or written, can the 



PEEFACE 5 

bookwork be learnt. There are a few sets of oral 
examples; these are intended to fill up spare minutes 
at the end of a lesson and often bring out the weak 
points in a pupil's knowledge. The book contains nearly 
1,000 examples ; it is not intended that any one should 
attempt all these, but it is hoped that they include 
a sufficient variety of types and a sufficient number of 
each type to meet all requirements. 

Many examples are taken from Examination Papers 
by kind permission of the following authorities : — 

The Controller of His Majesty's Stationery Office. 

The University of Cambridge. 

The Joint Matriculation Board of the Scottish Univer- 
sities. 

The Intermediate Education Board for Ireland. 

The Oxford and Cambridge Schools Examination 
Board. 

The Delegacy for Oxford Local Examinations. 

The Syndicate for Cambridge Local Examinations. 

The College of Preceptors. 

I am indebted to Mr. Norman Chignell, B.A., of 
Charterhouse, for many suggestions and for assistance 
in correcting the proof-sheets. It is too much to hope 
that the answers are wholly free from mistakes, and 
I shall be grateful to receive early intimation of any 
corrections that may be found necessary. 

VV. E. P. 

April, 1911. 



CONTENTS 



CHAP. PAGE 

Preliminary. Propositions in Geometry. Graphs. 

Logarithms. Greek Alphabet ... 9 

I. Angles and their Measurement. The Degree. 
Positive and Negative Angles. The 
Grade. Connexion between Arc and 
Angle. Circular Measure. The Points 
OF THE Compass. Gradient .... 25 

II, Definitions of Trigonometrical Ratios. Their 
Graphs. Inverse Notation. Construc- 
tion OF Angle with given Ratio . . 37 

IIT. Elementary Formulae. Identities. Ratios 
OF Complementary and Supplementary 
Angles. Ratios of 0^ 90", 30°, 60°, 45^ 

The Small Angle 49 

Revision Examples A .65 

TV. Use of Tables. Harder Graphs. General 
Solutions of Equations. Use of Loga- 
rithms 70 

V. The Right-angled Triangle Definitions of the 
Trigonometrical Ratios. Solution of 
Right-angled Triangles. Navigation: 
Plane Sailing. Ratios of the Double 
Angle and Half Angle 85 

VI. Relations between the Sides and Angles of a 
Triangle. Area of Triangle. The Circles 
of the Triangle 107 

Vn. Solution of Triangles 119 

Revision Examples B 134 



3,.^., CONTENTS 

CHAP. PAGE 

VIII. Projection. Vector Quantities. Composition 
AND Resolution of Vectors. Projection 
OF AN Area on a Plane. Geometrical 
Representation of Imaginary Quantities. 
Demoivre's Theorem 139 

IX. Ratio of the Sum or Difference of two Angles. 
Sum or Difference of Two Sines or Co- 
sines. Product of Sines and Cosines. 
The Double Angle and Half-Angle . .155 

Revision Examples C 175 

Miscellaneous Problems (taken from recent 
Army Entrance and Civil Service Exami- 
nation Papers) 184 

Examination Papers 190 

INDEX 201 

The following course of reading is suggested for l3eginners : — 
Chapter I, §§ 1-4, 9-11. 
Chapter II. 
Chapter III, §§ 21-31. 
Chapter IV, §§ 34-7, 40-2. 
Chapter V, §§ 43-5, 51-3. 
Chapter VI, §§ 54-8. 
Chapter VII. 



PliELIMINAEY CHAPTEE 

GEOMETRY 

A KNOWLEDGE of the following geometrical facts is required.* 
In this book these propositions are referred to by the numbers 
given below. 

Angles. 

Prop. 1. If a straight line meets another straight line, the 
adjacent angles are together equal to two right angles. 

Prop. 2. If two straight lines cut, the vertically opposite angles 
are equal. 

Prop. 3. The angle at the centre of a circle is double an angle 
at the circumference standing on the same arc. 

Prop. 4. Angles in the same segment of a circle are equal. 

Prop. 5. Angles at the centre of a circle standing on different 
arcs are in the same ratio as the lengths of the arcs. 

Triangles. 

Prop. 6. [a] The three angles of a triangle are together equal 
to two right angles. 

(&) If one side be produced the exterior angle equals the sum of 
the two interior opposite angles. 

Prop. 7. Any two sides of a triangle are together greater than 
the third. 

Proj). 8. Two triangles are congruent (i. e. are equal in every 
respect) if they have — 

(a) two sides of the one equal to two sides of the other, each 
to each, and the angle contained by the two sides of the one equal 
to the angle contained by the two corresponding sides of the other ; 

or (h) three sides of the one equal to three sides of the other, each 
to each ; 

or (c) two angles of the one equal to two angles of the other, 

* For proofs see Warren's Experimental and Theoretical Geometry 
(Clarendon Press), or any standard textbook. 



10 GEOMETRY 

each to each, and a side of the one equal to the corresponding side 
of the other. 

Prop. 9. If two triangles have an angle of the one equal to 
an angle of the other, and the sides about another pair of angles 
equal, each to each, then the third angles are either equal or 
supplementary. 

Prop. 10. (a) If two sides of a triangle are equal, the opposite 
angles are equal. 

(&) If two sides are unequal, the greater side is opposite a greater 
angle. 

(c) If all the sides of a triangle are equal, all the angles are equal. 

Prop. 11. (a) If two angles of a triangle are equal, the opposite 
sides are equal. 

{b) If two angles are unequal, the greater angle is opposite a 
greater side. 

(c) If all the angles of a triangle are equal, all the sides are equal. 

Prop. 12. Two triangles are similar (i. e. their angles are equal, 
each to each, and the ratio of pairs of sides opposite equal angles 
is the same for all three angles) if they have — 

{a) their angles equal each to each ; 

(6) their sides in the same ratio ; 

(c) an angle of the one equal to an angle of the other, and the 
sides about the equal angles in the same ratio. 

Prop. 13. (Pythagoras' Theorem.) In a right-angled triangle the 
square on the hypotenuse is equal to the sum of the squares on the 
other two sides. 

Parallel Lines. 

Prop. 14. (a) If a line is drawn to cut two parallel lines, it 
makes (i) the alternate angles equal, (ii) the interior angles on 
the same side of it together equal to two right angles, (iii) the 
exterior angle equal to the interior opposite angle. 

(b) The opposite sides and angles of a parallelogram are equal. 

Area. 

The unit of area is the area of a square whose side is of unit 
length. 

Prop. 15. The number of units of area in a rectangle is equal to 
the product of the number of units of length in one side multi- 
plied by the number of units of length in the other. 



GEOMETRY 11 

Or, more shortly : Area of rectangle = length x breadth. 
Proio. 16. The area of a triangle = \ base x altitude. 

The Concurrencies of the Triangle. 

Prop. 17. The lines bisecting the sides of a triangle at right 
angles are concurrent (i. e. meet at a point). 

The point in which they meet is the centre of the circle passing 
through the three vertices and is called the circumcentre. 

Prop. 18. The lines drawn from the vertices to bisect the oppo- 
site sides are concurrent. 

These lines are called the medians and the point of concurrency 
is called the centroid. 

Prop. 19. (a) The lines bisecting the angles are concurrent. 

The point of concurrency is the centre of the circle that touches 
all the sides, and is called the incentre. 

(&) If two of the sides be produced, the lines bisecting the ex- 
terior angles so formed and the line bisecting the interior angle 
contained by the produced sides are concurrent. 

The point of concurrency is the centre of the circle that touches 
the two sides when produced and the third side (not produced) ; 
it is called an e-centre. 

Prop. 20. The perpendiculars let fall from the vertices on the 
opposite sides are concurrent. 

The point of concurrency is called orthocentre. 

The Circle. 

Prop. 21. The straight line passing through the centre, at right 
angles to a chord, bisects the chord. 

Prop. 22. [a) The angle at the centre of a circle is twice the 
angle at the circumference on the same arc. (&) Angles in the 
same segment are equal, (c) The opposite angles of a quadri- 
lateral inscribed in a circle are together equal to two right angles. 

Prop. 23. The tangent at any point is at right angles to the 
radius drawn to that point. 

Prop. 24. (a) Two tangents can be drawn to a circle from any 
external point. (&) The parts of these tangents between the 
external point and the points of contact are equal, (c) The line 
joining the external point to the centre bisects the angle between 
the tangrents. 



12 



GEOMETRY 



Prop. 25. The ratio of the circumference of any circle to its 
diameter is the same for ail circles. 

This ratio is denoted by the symbol tv : its value is 3*1416 correct 
to five significant figures. 

Prop. 26. The area of a circle equals the area of the rectangle 
contained by the radius and a straight line equal to half the 
circumference. 

This is usually expressed in the formula : Area = tt r^. 

GRAPHS* 

Geometrical. If two straight lines are drawn in a plane, the 
position of any point in the plane can be determined by means of 
its distances from those lines. 



Y 

Fig. 1. 

It is usual to draw one of the lines horizontal and the other 
perpendicular to it. The customary notation is shown in Fig. 1. 

X'OX is called the axis of x ; 

Y'OY i% called the axis of y; 

is called the origin ; 

ON is called the abscissa of the point P; 

NP is called the ordinate of the point P. 

The abscissa and ordinate are called the co-ordinates t of the 
point P. 

* For a fuller treatment of Graphs see School Algebra published by 
the Clarendon Press. 

t Tliese co-ordinates are called Cartesian co-ordinates because they 
were first used by the French mathematician, Descartes. 



GRAPHS 



13 



The abscissa is said to be positive if drawn to the right, negative 
if drawn to the left. Similarly, the ordinate is positive if drawn 
upwards from N, negative if drawn downwards. The number of 
units of length in ON, preceded by the proper sign, is usually 
denoted by x, and the number of units of length in NP, preceded 
by the proper sign, is denoted by tj. In each case the sign + is 
often omitted. 

Thus, in Fig. 2, the co-ordinates of A are x = -4, y = 2, of 
C, x = 4:, y = ^, oi L, x = (), y = -7. 











1 












1 1 








1 1 








1 
























Y 






















































































































































io 
















1 
































































































































B 






















































5 
































































1 
















































ID 




















1 


















c 


















X 














A 














































































X 






















O 




r~ 








5 








K 








































































1 














1 










1 


















Gl 
















E 
























H 






















































5 


. 


























i 1 










1 




1 


V 
















1 


























LI 
















































1 
















































1 






















1 
























^°, 










1 










1 
























1 










i 


































1 








1 
































Y| 1 


1 


1 1 






i 



Fig. 2. 

Very often a point is described by writing the values of the 
co-ordinates in brackets ; e. g. the point H might be described as 
the point ( — 6, —4). 

Exercise. Write down the co-ordinates of all the points in 
Fig.- 2. 

Graphs of Statistics. The magnitude of any quantity may be 
represented by a straight line which contains as many units of 
length as the quantity contains units of its own kind. 

If two quantities are changing their values at the same time, 



14 



GRAPHS 



the simultaneous values may be represented in the same figure 
by taking horizontal lengths to represent one magnitude and 
vertical lengths to represent the other. 





1 


























~ 






5 



























1 


' 
































1 


































1 
































i 


































/ 










4 


5 






















/ 


































/ 


































' 
























































\ 






















4 











1 


\ 


































\ 






























/ 




\ 






























/ 


































/ 


























3 


5 




1 








\ 


























/ 














1 














i3 




















/ 












« 


3> 




/ 
















/ 












c 


i 




1 
















1 
















O 


















/ 
































V 


/ 


















































































j 1 




\ 




1 
















1 


S.\M. 


T. 


W. 1 Th. 


F. 


Sot. 




1 


i i 1 


1 


1 1 1 




1 





Fig. 3. 



In Fig, 3 the changing quantities are time and temperature ; 
and the dots show that at noon on Sunday the temperature was 
29°, on Monday the temperature was 35, &c. In fact the figure 
conveys the same information as the following table : — 

Sun. Mon. Tues. Wed. Thurs. Fri. Sat. 
Temp. 29° 85° 42° 31° 27° 43° 50° 

If there is no information about intermediate temperatures, 
the points are joined by a series of straight lines. The figure 
now forms a graph. 

In describing such a graph we should say that the abscissae 
represent time and the ordinates temperature. 

Graphs of functions. If two quantities x and // are such that 
a change of value in the one causes a change of value in tiie 
other, then either of them is said to be a function of the other. 



GRAPHS 



15 



This is expressed thus : y =f{x), or x =f{y) where f[x) means 
a function of x. A graph can be drawn in which the abscissae 
are proportional to the values of x and the ordinates to the values 
of y. This graph is called the graph of the function f{x) or of 
the equation y=f{x). This maybe more easily understood by 
considering a few algebraical functions. 

Example I. Draw the graph when y = \x-% 

(Choose values of x which will make y a whole number.) 

y = \{^x-2) 

X -6-149 

Zx-1 -20 -5 10 25 

y _ 4 -1 2 5 



- 
















































































































































1 















































































































- 




















































































f^ 


























5 














y 


^ 








































y 


































! 




/ 


r^ 


































1 I>^1 1 
































^ ! ' 























5 






u 


"51 




"l 


O 
















^ 


f" 














- 






! 






1/ 








1 ' ' ' 1 


















y 






1 ! 


: I 1 




















/^ 


y 












1 


1 




















k- 
















5 






































i 






































1 






1 

















Fig. 4. 



When the points corresponding to these values of x and y are 
plotted, it is found that they lie on the straight line shown in 
Fig. 4, It is also found 

(i) That any simultaneous values of x and y connected by the 
given equation are the co-ordinates of some point on this straight 
line ; 

(ii) That the co-oidinates of any point on the straight line 
satisfy the equation. 

It is found by experience (and can be proved from the geo- 



16 



GRAPHS 



metrical propositions on proportion) that, when x and y are 
connected by an equation of the first degree, the graph is always 
a straight line. 

Example II. In the same figure draw the graphs of 

y = x^-^x-V^ and r» = 2f/H3. 
Neither of these- equations is of the first degree, therefore 
neither of the graphs is a straight line. At least six points must 
be found on each. 

yr=x^-Zx\'l. (i) 



X 


-3 


— 2 


-1 





1 


2 


3 


4 


5 


x^ 


9 


4 


1 





1 


4 


9 


16 


25 


-3a; 


9 


6 


3 
6 




2 


-3 


-6 


-9 


-12 
6 


-15 
12 


y 


20 


12 








2 



JT 20 


f I 


t -1 


X II 


4 3f 


4 ^ 4t 




1 






:: ri i 


-L \ J 


i f: 




u 3 


l" 7 


4 A =+ 


r _ — — =-^ 


j^^-;^" 


-- -^ S^Z 


5 ^ i ^ 5 lO 5 


^ -- — -___ 









Fig. 5. 



GRAPHS 



17 

(ii) 



y 


-3 


-2 


-1 





1 


2 


3 


4 


2j/2 


18 


8 


2 



-3 


2 


8 


18 


32 


X 


15 


5 


-1 


-1 


5 


15 


29 



The co-ordinates of every point on graph i satisfy the first equa- 
tion, and of every point on graph ii satisfy the second equation. 
Hence the co-ordinates of any points which are on both graphs, 
that is, the co-ordinates of the points of intersection, satisfy both 
equations. 

Fig. 5 shows, therefore, that the values x = '3, t/ = I'S, and 
X = 2'8, y = 1*6, are the solutions of the two equations. 

This graphical method of solving equations is very useful, 
but is, of course, only approximate. If more accurate answers are 
required, the graphs must be drawn on a larger scale in the 
neighbourhood of their points of intersection. 

LOGARITHMS * 

Fractional and Negative Indices. It is shown in Algebra 
that p _ 1 

where p and g are any positive integers and x is any positive 
quantity, integral or fractional. 

A fractional index may be expressed as a decimal ; thus such 
expressions as 4'^^ 10*'°^ have a definite value. This value could 
in theory be found by reducing the decimal to a vulgar fraction 
and then replacing the power with a fractional index by a root, 
e.g. 

10-301 

This is obviously not practical. The value can be found by a 
graphical method which is easy but only approximate. 

Di-aw the graph of x = lO*'. 

* For a fuller treatment of Indices and Logarithms see School Algebra, 
Chapters XXI and XXII. 



_301_ , 

IQIOOO _ 1000 /j^QSOl 



\ 



18 



LOGARITHMS 



y 





•5 


•25 


•125 


'lb 


•625 


•875 


1 


X 


1 


3-16 


1-76 


1-33 


5-62 


426 


7-49 


10 



The values of x are obtained as follows : — 
10° = 1 by definition given above. 
10-» = 10^ = VIo. 

10-25 = 10^= V^\/ 10; similarly W^'-^^^ ^W\ 
W^ = 10-5+-25 = 10-5 X 10*25, &c. 

The graph is shown on a small scale in Fig. 6. 

It is seen that 10'^° is almost exactly 2 ; any other power of 10 
can be found approximately from this graph when the index is 
between and 10. 

Definition of a logarithm. The logarithm of a number to a 
given base is the index of the power to which the base must be 
raised to equal the number. Thus 3^'= 9, therefore the logarithm 
of 9 to base 3 equals 2 ; this is written log3 9 = 2. 

In dealing with numbers the base is 10. In the remainder of 
this chapter it is assumed that the base is always 10, so that 
log 731 means logarithm of 731 to base 10. 

The equation x = 10^ may be written t/ = log x. 

Hence Fig. 6 provides an approximate means of finding the 
logarithm of any number between 1 and 10. 

Characteristic and mantissa. Consider a number, such as 
4878. It means 

4xlO.H3x 102 + 7x10 + 8. 
Also a decimal number, such as '0376, means 

A A _I_ _?_ 
10 "^ 1Q2 "^ 10^ "^10^* 

If we use negative indices, this may be written 

•0376 = 3 X 10-2 + 7 X 10-^ + 6 x 10"*. 
Similarly 

537-13 = 5 X 102 + 3 x 10^ + 7 X 10'' + 1 X 10-i + 3 x lO"''. 
It follows that 

4378 > 103 but < 10\ 

•0376 > 10-2 but < 10-^ 

587-13 > 102 but < 10», 

4-37 > 10° but < 10\ 



•. log 4378 = 3 + a decimal ; 
•. log "0376 = - 2 + a decimal ; 
•. log 537-13 = 2 + a decimal ; 
•. log 4'37 = + a decimal. 



LOGARITHMS 



19 



We now see (i) that the logarithm of any number consists of an 
integer (which may be positive, zero, or negative) and a positive 




Fig. 6. 

decimal, (ii) that the integer is the index of the highest power of 
10 contained in the number. 

The integral part of a logarithm is called the characteristic ; 
the decimal part the mantissa. 



Index 


5 


, 


3 


2 


1 





-1 


-2 


-3 


-4 


-5 


-6 


Number 


3 


7 


8 


9 




















)j 






4 


3 


7 


6- 


5 


2 










?> 












3- 


4 


6 


7 


8 






»> 












0- 








7 


3 


4 


8 


if 






2 








0- 


















20 



LOGAEITHMS 



Consideration of the preceding table shows that the character- 
istic (i. e. the highest index) may always be found by the following 
rule : Count from the unit place to the first significant figure 
(i. e. the first figure which is not 0), the unit place being counted 
as nothing. The characteristic is positive or zero if the number 
is greater than one, negative if it is less than one. 

The mantissa is independent of the position of the de- 
cimal point. An example will make this clear. 

Given that log 4-376 = -6411, find log 4376 and log -004376. 
4376 = 1000 X 4-376 

but 4-376 = 10-"" since log 4*376 = '6411. 
.-. 4376 = 10^ X 10-<'^^i = 10s-6"S i. e. log 4376 = 3-6411 
•004376 = 1 J-oo X 4*376 = 10-^ x lO-^^i = io-3+-64n 
.-. log -004376 = -3 + -6411. 

The negative sign of a characteristic is always placed on top 
and the + before the decimal is omitted. Thus log '004376 = 
3-6411. 

To find the logarithm of any number. 

(a) Four-figure tables. The mantissa is found from tables, of 
which a specimen is given below. 

Logarithms. 








1 


2 


3 


4 


5 


6 


7 


8 


9 


I 2 


3 4 


5 


6 7 8 


51 

53 
54 


7076 
7160 
7243 
7324 


7084 
7168 
7251 
7332 


7093 
7177 

7259 
7340 


7101 
7185 
7267 
7348 


7110 
7193 
7275 

7356 


7118 
7202 
7284 
7364 


7126 
7210 
7292* 
7372. 


7135 
7218 
7300 
7380 


7143 
7226 
7308 
7388 


7152 
723s 
7316 
7396 


I 2 
I 2 
I 2 
I 2 


3 3 
2 3 
2 3 
2 3 




' i 7 

5 ^ I 
566 

5 <i 6 


55 


7404 


74r2j 


7419 


7427 


7435 


7443 


7451 


7459 


7466 7474 


I 2 


2 3 




5 5 6 



Consider the logarithms of 5467 and '05467. 

By counting, the characteristic of log 5467 is found to be 3, and 
that of log '05467 to be 2. 

Both logarithms have the same mantissa. Look for 54 in t)ie 
extreme left-hand column. In the same line with 54 and under 6 
we find 7372 ; this is the mantissa of log 546. Under the 7 in the 
small columns to the right, and in a line with 54, we find 6 ; this 
must be added to the last digit of the mantissa already found. 
Hence the mantissa is '7378. 

Therefore log 5467 = 3'7378 and log '05467 = 2*7378. 



LOGARITHMS 



21 



(b) Five-figure tables. Find the logarithm of 346'73. 

Proceeding as -with four-figure tables, we find that the mantissa 
of log 346 is '53908. ,Under 7 in the side columns, we find 88 ; 
this must be added to the last two digits already found. For a 3 
in the fourth place we should add 38, for a 3 in the fifth place we 
add, therefore, ^^ x 38, i. e. 4 to nearest integer. Hence 
log 3467-3 = 2-53908 + 88 + 4 = 2-54000. 

To find the number corresponding to any logarithm. 

Method I. Reverse the process for finding a logarithm. Sup- 
pose the logarithm is 3*7271. 

Look in the logarithms for the mantissa nearest to 7271, but 
less than it. We find 7267, level with 53 and under 3 ; the first 
three figures of the number are 533. This leaves 7271 - 7267 = 4 ; 
in the right-hand columns 4 is found under 5. Hence the first four 
figures are 5335. 

The characteristic is 3, therefore the left-hand digit 5 represents 
5 X 10^ ; hence the number is bd\p. 

The number is called the antilogarithm of the logarithm. 

Method II. If tables of antilogarithms are available, they are 
used in the same way as kcarithm tables. 



AxmLOGARITHMS. 






1 


2 


3 


4 


5 


6 


7 


8 





12 3 4 


5 


6 7 8 9 


-012 


5023 


5035 


5047 
5164 
5284 
5408 

5534 
5662 


3058 
3176 
5297 
5420 
5546 
3675 


5070 


508^ 
5200 
3321 

3445 
5572 
5702 


5093 
5212 
5333 
3458 
5585 
5715 


5105 
3224 
3346 
5470 
5598 
5728 


5117 
5236 
3358 
5483 
5610 

5741 


1245 


6 


7 8 9 n 


5370 

5493 


5140 
5260 
5383 

5508 


5152 
5272 

5393 
3521 


3188 
3309 

3433 
3559 


1245 
1245 
1345 
1345 


6 
6 
6 
6 


7 8 10 II 

7 9 10 II 

8 9 lo II 

8 9 10 13 


5623 


5636 


3649 


5689 


1343 


7 


8 9 10 12 



Look for *72 in the left-hand column ; level with *72 and under 7 
we find 5333 ; in the small columns we find 1 under 1. Hence the 
first four figures are 5334. 

The characteristic is 3 ; as before, the number is 5334. 



Note. — The two methods give results differing by 1 in the last 
figure ; this shows that the number is between the two results. On 
using five-figure tables, it is found that the antilogarithm of 3'7271 
is 5334-5. 



22 LOGARITHMS 

Use of Logarithms. 

By the definition of logaritlim a = lO^oga, & = lO^os^, 

.-. db = 10log«x IQlos^ = iQlogrt+logft. 
Hence log (a6) = loga + log &. 

Similarly log ajh = log a — log &, 

log a'" = wi log a, 

log Vi/a = — log a. 

Thus, instead of 

multiplying, we may use logarithms and add ; 
dividing, ,, „ subtract; 

raising to a power, „ ,, multiply ; 

taking a root, „ „ divide. 

Note. — There is no process with logarithms to correspond with 
addition or subtraction with ordinary numbers. 

1 T 17- ^ M 1 P 516-5 X -852 
Example I. Find the value of —^- — • 

log of fraction = log SIG'S + log '852 -log 36500 
= 2*7130 
+ 1-9304 -4-5623 
= 2-6434 
- 4-5623 
= 2-0811 
.-. fraction = -01205. 

Example II. Find the cube root of -1765. 
log of cube root = ^ log '1765 
= ^ of 1-2467 
= ^3 + 2-2467) 
= 1-7489. 

Notice carefully this method of division when the characteristic 
is negative. 

Hence V-1765 = -5610. 





LOGARITHMS 


Exercises. Find the value 


of 




(1) 
(2) 
(3) 
(4) 
(5) 


^^319-2 X r756. 
•03056 X 0-4105. 
3-142 x(71-43)^ 
^x 3-142 X (9-67)3. 
254-3 -r 0-09027. 


Ans. 
Ans. 
Ans. 
Ans. 
Ans, 


31-37. 

•01254. 

16030. 

3787. 

2817. 


(6) 


(-1136)^x^/81-86 
^x V2000 


An?. 


•1874. 



23 



THE GREEK ALPHABET 

Greek letters are used so frequently in Trigonometry and other 
branches of Mathematics that it is useful to have the complete 
alphabet for reference. 



Name. 


Small. 


Capital. 


alpha 


a 


A 


beta 


^ 


B 


gamma 


y 


r 


delta 


d 


A 


epsilon 


e 


E 


zeta 


c 


Z 


eta 


V 


H 


theta 


e 


e 


iota 


L 


I 


kappa 


K 


^ : 


lambda 


X 


mu 


/* 


M 


nu 


V 


N 


xi 


i 


S . 


omicron 


o 


O 


pi 


IT 


n 


rho 


P 


p 


sigma 


a- 


2 


tau 


T 


T 


upsilon 


V 


Y 


phi 





* 


chi 


X 


X 


psi 


^ 


^ 


omega 


<o 


Q« 



CHAPTER I 

ANGLES AXD THEIR MEASUREMENT 

1. Any angle such as BAC (Fig. I) may be thought of as 
having been formed by rotating the line AC about the point A 
from the position of coincidence with AB to its final position AC. 




Fig. I. 

This way of regarding an angle shows clearly the intimate 
connexion between angles and arcs of circles and this connexion 
leads to the usual method of measuring angles. 

2. The Degree.* From very early times it has been the custom 
to divide the circumference of a circle into 360 equal parts or 

* 'The current sexagesimal division of angles is derived from the 
Babylonians through the Greeks. The Babylonian unit angle was the 
angle of an equilateral triangle; following their usual practice this 
was divided into 60 equal parts or degrees, a degree was subdivided 
into 60 equal parts or minutes, and so on ; it is said that 60 was 
assumed as the base of the system in order that the number of degrees 
corresponding to the circumference of a circle should be the same as 
the number of days in a year which it is alleged was taken, at any rate 
in practice, to be 360.' (From A Short Account of the Hisfory of Mathe- 
matics, by W. W. Rouse Ball.) 



26 ANGLES AND THEIR MEASUREMENT 

degrees, each degree into 60 parts or minutes,* each minute into 
60 seconds.^ 

The angle at the centre of a circle, subtended by an arc 
of 1 degree, is taken as the unit angle, and it, too, is called 
a degree ; it is divided into minutes and seconds in the 
same way as the arc degree. 

The notation used is shown in the following example : — 
47° 15' 37" is read 47 degrees 15 minutes 37 seconds. 

If the line makes a complete rotation, thus returning to its 
original position, it has turned through an angle of 360°. 

A right angle is produced by one-quarter of a complete rotation, 
and is, therefore, equal to 90°. 

If two angles together equal a right angle, either of them is 
called the complement of the other. When the sum equals two 
right angles, either angle is the supplement of the other. 

3. Positive and Negative Angles. In discussing the 
properties of a single angle it is usual to draw the initial 
line so that it is horizontal and to name it OA. If the 
rotating line moves in a direction opposite to that of the 
hands of a clock, the angle is said to be positive ; if in 
the same direction as the hands of a clock, the angle is 
negative. 




Fi- III. 



* Minutes derived from the Latin jmrfcs minutae ; seconds from the 
Latin partes fuinntae secundac. 



ANGLES AND THEIR MEASUREMENT 27 

In Fig. II the line OP has made i of a complete turn, hence 
the angle ^OP = i x 360° = 45° ; in Fig. Ill the angle ^ OP is 
reflex * and is equal to § x 360° = 225°. If, in Fig. II, the line 
OP reached its position b}-- turning in the negative direction it 
would have made | of a complete turn so that the reflex angle 
^OP in Fig. II = -315°. Similarly in Fig. Ill the obtuse angle 
AOP= -135°. 

4. Angles unlimited in size. In Fig. II the line OP might 
have made one, two, or any number of complete turns, either 
positive or negative, and then have moved on to its final position : 
hence the angle AOP may represent 405°, or 765°, or —675°. All 
possible values are included in the general formula 

^0P= 360 n + 45, 
where n is any whole number, positive, zero, or negative. 

Unless the problem under discussion allows the possibility of 
the angle being greater than 360°, it is always assumed that the 
angle is less than 360°. 

5. The Grade. When the metric system was invented, the 
French Mathematicians introduced a new unit, the Grade, such that 

100 grades = 1 right angle, 

100 minutes = 1 grade, 

100 seconds = 1 minute. 
This system never came into general use, even in France, and now 
exists only in old-fashioned examination papers. 

Examples I a. 

1. Find the complement of each of the following angles: 
32°, 47° 23', 75° 13' 14", 68°0'13", 27° 42' 18-6". 

2. Write down the supplements of 75°, 68° 14', 115° 17' 48" 
90°, 78° 24' 36". 

3. The angles of a triangle are found to be 42° 13' 17", 73° 47' 5", 
64° 0' 38". Is this correct ? 

4. Two angles of a triangle are 17° 43', 92° 16'; calculate the 
third angle. 

5. In a triangle ABC, 1{A + B) = 77° 29' and \{A-B) = 16° 25' ; 
find all the angles. 

* A reflex angle is an angle greater than two right angles, but less 
than four right angles. 



28 ANGLES AND THEIR MEASUREMENT 

6. Express in degrees, minutes, and seconds the angle of 
(a) a square, {h) a regular pentagon, (c) a regular heptagon. 

7. Express each of the angles of question 6 in grades. 

8. The magnitude of an angle may be expressed either as 
D degrees or G grades ; find the equation connecting D and G. 

9. Draw the angles A and ^A in each of the following cases: 
(«) A = 54°, (b) A = 414°, (c) A = 774°, (d) A = 1134°, (e) 234^ 
(/) -126°. 

10. Through what angles do the hour, minute, and second 
hands of a watch respectively turn between 12^ 30' a.m. and 
5h 3' a.m. ? 

6. The ratio of the length of the circumference of a circle 
to the length of its diameter is the same for all circles. 
This constant value is denoted by the Greek letter -n (pro- 
nounced pi), so that if 

the circumference = c units of length, 
and diameter = cl units of length, 

then - = 77. 

d 

The value of tt can be found, correct to two or three significant 
figures, by actual measurement. By geometrical and trigono- 
metrical calculations its value can be calculated to any desired 
number of places. 

Correct to 5 significant figures, n = 31416. 

Correct to 6 significant figures, n = 314159. 

For mental calculations tt may be taken as 31. 

7. By using Prop. 5, p. 9, problems dealing with the lengths of 
circular arcs may often be solved. 

Example. Find the length of an arc which 
subtends an angle of 49° at the centre of a 
circle whose 7-adius is 5 feet. 




arcylP _ angled OP 

semi-circumference 2 right angles 
ai'c AP __ 49° 
Fig. IV. i-^- "^5^ ~180°* 

The calculation is easily completed. 



ANGLES AND THEIR MEASUREMENT 29 

,,. ., , area of sector ^OP an^le ^OP ,t^ o^\ 

Similarly r^l = , .^, . 1— , (Pi'op. 26) 

"^ area of circle 4 rierht angles 



1. e. 



area of sector AOP _ 49 
2blT 360 



8. Circular measure. By the method of the last section 
it is easily shown that the length of an arc of a circle, 

Att 

radius r, subtending an angle A° at the centre is r -— - . 

loO 

In many other formulae the fraction -— occurs in con- 

nexion with the angle A°. In theoretical work it has, 
therefore, been found convenient to use another unit angle, 
which simplifies formulae considerably. 

The radian is the angle subtended at the centre of any 
circle by an arc equal in length to the radius. 

Let x° equal 1 radian 

arc equal to radius _ angle of 1 radian _ x°^ 
semi-circumference 2 right angles 180° 

'•^* 7rr~180°' 

Since n is the same for all circles, it follows that the radian is 
the same for all circles and may, therefore, be taken as a unit 
of measurement. 

The number of radians in an angle is often called the 
circular measure of the angle. For this reason the 
symbol <^ is used to show that the angle is measured in 
radians, e. g. 2^^ means 2 radians. 

When the radian is the unit angle, it is customary to use 
Greek letters to denote the number of radians, and the 
symbol '^ is then often omitted. When capital English 
letters are used, it is usually understood that the angle is 
measured in degrees. 



30 ANGLES AND THEIR MEASUREMENT 

Examples I b. 

1. How many times is an arc equal to the radius contained in 
the semi-circumference ? Reduce 180°, 90°, 60°, 30° to radians. 
(Do not substitute for tt.) 

2. Show by simple geometry that the radian is less than 60°. 

3. How many radians are there in 10°, 75°, 138°, respectively ? 
Give the answers correct to 2 decimal places. 

4. Express the angle of (i) an isosceles right-angled triangle, 
(ii) a regular nonagon, in circular measure. Give the answers in 
terms of 73-. 

5. One angle of a triangle is ^ tt, another is -J- n ; what is the 
circular measure of the third angle ? 

6. Find the length of an arc of a circle which subtends an 
angle 78° at the centre, the radius being 18 feet. 

7. An arc of length 5 feet subtends an angle of 132° at the 
centre ; what is the radius of the circle ? 

8. Find the area of the sector of a circle if the radius is 12 feet 
and the angle 40°. 

9. What time does the minute hand of a watch take to turn 
through (i) 3000°, (ii) 3000 grades, (iii) 3000 radians ? 

10. Fill in the missing values in the following table, which gives 
data about circular arcs. 

Length. 



413 feet 
220 yards 
1 kilometre 
half a mile 



11. Express in radians the angle of a sector of a circle, being 
given that the radius is 7 inches and the area of the sector 100 sq. 
inches. 

12. Show that the length of an arc subtending an angle 0^ at 
the centre of a circle, radius r, is r6. What is the area of the 
corresponding sector ? 

13. Find the circular measure of V and of \'\ correct to 5 signi- 
ficant figures. 





Kadius. 


Angle. 


(1) 


5 inches 


2 radians 


(2) 


7-6 centimetres 


74*6 grades 


(3) 




314 degrees 


(4) 


100 yards 


radians 


(5) 


320 metres 


degrees 


(6) 


yards 


5 radians 



ANGLES AND THEIR MEASUREMENT 



31 



9. The points of the compass. The card of the Mariner's 
Compass is divided into four quadrants by two diameters pointing 
North and South, East and AVest respectively. These are the 
Cardinal Points. Two other diameters bisecting the angles 
between the previous diameters give four other points, viz. NE., 
NW., SW., SE. The eight angles so formed are bisected and 




eight more points are thus obtained. These are named by com- 
-biningthe names of the points between which they lie, beginning 
with the cardinal point. Thus the point midway between E. and 
SE. is ESE. (East South-East). 

The sixteen angles now formed are bisected so that the circum- 
ference is finally divided into thirty two equal divisions. From 
their names the last sixteen points are called by-points. The 
point midway between N. and NNE. is called N. by E. ; that mid- 
way between SW. and SSW. is called SW. by S., &c. 



32 



ANGLES AND THEIR MEASUREMENT 



The angle between two consecutive points of the compass is also 
called a point, thus N. 2 points E. is the same as NNE. ; WSW. \ W. 
means | a point W. of WSW. 

The ordinary degree is sometimes used in defining a direction, 
for instance ENE. can be referred to as 221° N. of E. Similarly 
we may have 32° W. of N., 40° S. of W., &c. 



10. Latitude and Longitude. The position of a point on 
a sphere can be defined by two angles, which may be compared 
with the abscissa and ordinate of plane geometry. These angles 
are easily understood by considering the special case of Longitude 
and Latitude. 




In Fig. V the meridian through Greenwich cuts the equator 
at A ; the meridian through P cuts at B. is the centre of the 
Earth. 

The Longitude of Pis the angle AOB and may be either East 
or West of the Greenwich meridian. 

The Latitude of Pis the angle FOB and may be either North 
or South of the Equator. 

Note. — A geographical or nautical mile is the lengtli of an arc of a 
meridian (or of the equator) subtending an angle of 1' at the centre of 
the earth. 



ANGLES AND THEIR MEASUREMENT 33 

A ship travelling at the rate of I nautioal mile per hour is said to 
have a speed of one knot. 

11. Gradient. It is usual to estimate the inclination to 
the horizontal of a road or hill by the distance risen veitically for 
a certain horizontal distance. Thus a hill might be said to rise 
3 in 5 ; this would mean that if a horizontal line were drawn 
through a point B on the hill to meet the vertical line through 
a lower point A at C, then AC/BC would equal ?. The hill is said 
to have a gradient or slope of 3 in 5. 




Fig. VI. 

It is clear that in many cases it is easier to measure AB than BC; 
and some books take a gradient of 3 in o to mean a rise of 3 vertically 
for a distance 5 measured along the incline ; so that in the figiire 
AC/AB would be f. This latter interpretation of gradient is very 
common in books on Theoretical Mechanics. 

If the inclination is small, it makes no practical difference whicli 
interpretation of gradient is taken. 

It should be noticed that the angle is the same whatever units 
be used ; that is whether we consider a rise of 3 inches in 5 inches, 
3 furlongs in 5 furlongs, 3 miles in '5 miles. This follows from 
Prop. 12 0. 

Examples I c. 

1. Express in degrees the angle between 
(a) NNE. and E. by N. ; (/) S. 2 points W. and W. 2 points S. ; 
(&) W. by S. and SE. by N. ; (r/) 40° N. of W. and 30° E. of S. ; 

(c) ESE. and NE. by N. ; {h) NE. by E. and 1 point W. of N. ; 

(d) NNW. and SSE. ; (/) 30° S. of W. and ESE. ; 

(e) N. by W. and SW. ; (k) S. 2 points W. and W. 2 points N. 
In the following questions take the radius of the Earth to be 

4000 miles. 



34 ANGLES AND THEIR MEASUREMENT 

2. Two places on the Equator are 300 miles apart, find the 
ditt'erence of their Longitudes. 

3. Quito (Longitude 79' W.) and Macapa (Longitude SIJt W.) 
are both on the Equator, find the distance between them. What 
time is it at Macapa when it is noon at Quito ? 

4. Find the distance between Poole (Lat. 50° 43' N., Long. 
1°59' W.) and Berwick (Lat. 55° 46', Long. 1°59'W.). 

5. Find the distance between Cape Breton Island (Lat. 45° 50' N., 
Long. 60° W.) and the Falkland Isles (Lat. 51° 32' S., Long. 60° W.). 

Oral Questions. 

1. What is a degree ? How many degrees are there in the angle 
of a regular pentagon ? 

2. How big is each acute angle of an isosceles right-angled 
triangle ? 

3. One angle of a triangle is A°, another 30°, how big is the 
third angle ? 

4. What is meant by a negative angle ? When screwing an 
ordinary screw in, is the turning in the positive or negative 
direction ? 

5. Does the earth rotate in the positive or negative direction ? 
In which direction does the sun appear to move ? 

6. Do you usually draw a circle in the positive or negative 
direction ? 

7. The needle of a mariner's compass is deflected from its normal 
position through a positive angle 33i| degrees, to what point of the 
compass does it then point ? 

8. Express the following angles in circular measure : 90°, 00°, 
180°, 45°, 30°. (Give the answers in terms of n.) 

9. What is the locus of all places having latitude 35° N. V 

10. What is the locus of all places having longitude 15° W. V 

11. It is noon at the same time at two different places, what do 
you know about their longitudes or latitudes ? 

12. Give the latitude and longitude of the N. pole. 

Examples I. 

1. In a triangle ABC, A = 43° 15', B = 07° 38', calculate the 
number of degrees in (i) the angle C, (ii) the angle subtended at 



ANGLES AND THEIR MEASUREMENT 35 

the centre of the c ire urn circle by the side BC, (iiij the angle sub- 
tended at the centre of the inscribed circle by the side BC. 

2. Express in circular measure, correct to 3 significant figures, 
(a) the supplement of I'S? radians, (/;) 74°, [c) the angle of a regular 
octagon. 

8. Define a radian and a grade. If an angle, containing D 
degrees, may be expressed as either 6 radians or G grades, prove 
that 2)/180 = 6/tt = G/200. 

4. The hands of a clock are coincident at noon, through what 
angle does the hour hand turn before they next coincide ? 

5. Prove that whatever be the radius of a circle the size of the 
angle at the centre, which subtends an arc ec^ual to the radius, is 
constant. What is this angle called ? Show, by a geometrical 
construction, that it is a little less than 60°. 

6. A wheel of a cart is 4 feet in diameter, through what angle 
does it turn when the cart moves forward 10 feet? 

7. Explain how to find the length of a circular arc being 
given the number of degrees in the angle subtended at the centre 
and the length of the radius. 

^'. Two places on the p]quator differ in longitude by 87° 16', find 
the distance between them, correct to three significant figures. 
(Radius = 4000 miles.) 

9. Find the distance between a place, longitude 45° 17' E., lati- 
tude 0°, and another place, longitude 88°43'W., latitude 0°. 

10. Through what angle does the Earth turn between 9.30 a.m. 
and 4 p.m. ? 

11. When it is noon at Greenwich what time is it at (a) Calcutta 
(88° 15' E.j, (&) New York (74° W.), ic) Hawaii (156° W.) ? 

12. The co-ordinates of two points P and Q are (7, 8), (9, 11) 
respectively, find the gradient of the line PQ. 

13. Draw an angle AOP = 85^ in OP take 8 points P, Q, li such 
that OP = 1 inch, OQ = VI inch, OR = 2d inches. From P draw 
PH Sit right angles to OA, at Q draw QK at right angles to OQ, 
and from R let fall RL perpendicular to OA. -Measure OH, OK, 
OL and calculate, to 3 decimal places, the ratios OH/ OP, OQ/OK, 
OLjOR. Justify the result. 

14. Explain what is meant by a radian, and find how many 
degrees and minutes it contains. 

Express in degrees, and also in radians, the angle of a regular 
polygon of 100 sides. 

c2 



S(j ANGLES AND THEIK MEASUREMENT 

15. An explorer reaches a latitude of 87° 28' 4«". Find how 
many miles he is distant from the pole, assuming the earth to be 
a sphere whose circumference is 25000 miles. 

16. Find the gradient of a straight line joining two points 
whose co-ordinates are {x', y'} and (x", y"). Hence find the equation 
of the straisrbt line. 



CHAPTER II 

THE TRIGONOMETRICAL RATIOS 

12. Definitions. Let OA the initial line be taken as axis 
of X, the axis of// being perpendicuhir to it at ; in the final 



P 

./I 



N A 



\ 




Fi?. VI r. 



Figure VII 



position of the rotating line take any point P. 
shows four possible types of positions of OP. 

Let fall PX perpendicular to OA or OA produced, so that 
ON is the abscissa of P and PN the ordinate. Then 



as THE TRIGONOMETRICAL RATIOS 

NP ordinate y 



sine of AGP = 
cosine of AGP = 



GP radius r 

GN abscissa x 



GP radius r 



. . . ^^ NP ordinate y *\ a 

tangent of AGP = -—- = , — ^ = - • 7. a. 

GN abscissa x ^ 

These are the most important ratios ; the others are their 

reciprocals, viz. : 

G P radius r 



cosecant of AGP — 
secant of AGP — 
cotangent of AGP = 



NP ordinate y 
GP radius r 



GN abscissa x 
GN abscissa x 



NP ordinate y 

TWfollowing abbreviations are usually used: 

sin A instead of sine of AOP, 

cos A ,, ,. cosine of ^ OP, 

tan^ „ „ tangent of .4 OP, 

cosec A „ ,, cosecant of AOP. 

sec A ,, „ secant of J OP, 

cotan ^ ,, ,, cotangent of. f OP. 
Similarly, if ^OPismeasured in radians, sin 6, cos f/^ cosec \//, &c., 
are used. 

13. Trigonometry was developed by Arabian and Greek astrono- 
mers who based their work on the circular arc and not on the 
angle. In the Middle Ages this early mathematical work was 
translated into Latin, and so the present names of the nitio were 
derived. The following section shows the reasons for these names. 

14. Draw a circle with centre cutting the initial line at .1 and 
the perpendicular to it at B. 

Take a point P on the circumference of the circle. 

Draw the tangent at yl and product' OP to meet it at T. 

Draw PN perpendicular to ()A. 

AT wiiH called the tangent of the arc AP. 

OT, which cuts the circle, was called the secant of the arc AP. 



THE TRIGONOMETRICAL RATIOS 



39 



. NP was called the sine * of the arc AP. 

Clearly the lengths o^ AT, 02] AP change when the radius OA 
changes, even if the angle .lOP remain constant. 




But from similar triangles it is seen that, if the angle is constant, 
the ratios oi AT, OT, NP to the radius are also constant. Hence, 
as Trigonometry developed, it was seen to he advisable to divide hy 



* The word ' sine' is derived from the Latin sinus. If in Fig, VIII 
P^y be produced to meet the circumference at P', then PAP' resembles 
a bow (Latin arcus) of which PXP' is the string or chord (Latin chorda}. 
To use the bow, the string is pulled till JV touclies the bosom (Latin 
sinus^ ; hence PX is called the sine. NA is often called the sagitta of 
the arc. 



40 THE TRIGONOMETRICAL RATIOS 

the radius and to treat the subject as depending on the angle 
AOF rather than on the arc A P. Thus we have 

angle AOP = '—^. — (when the angle is measured in radians), 
" radius 

,^, sine of arc ^P iYP 

sin AOP = 



tan AOP = 
sec AOP =- 



radius OP 

tangent of arc AP _ AT _ yP 
radius OA ON 

secant of arc AP OT OP 



radius OA ON 

Now make a similar construction for the complementary arc BP. 
Then 
sine of the complement of the angle AOP 

__ sine of the complementary arc BP 

radius 
^ XP 
OP 
_ ON 

~ OP ' '' 

' Sine of the complement of ' was shortened into co-sine. Possibly 
' complementary sine ' was an intermediate stage. [Similarly, co- 
tangent and cosecant were derived. 

Since the values of the ratios depend on the values of the angle, 
the term Trigonometrical Functions is often used instead of 
Trigonometrical Ratios. Frequently the ratios are referred to as 
Circular Functions. 

15. Ratios rarely used, in Fig. VIII 

N A is called the vorsine (i. e. versed sine) of the are AV. 

X'B is called t])e coversine of the arc AP. 

AP (not joined in the figure) is the chord of the arc AP. 
If we divide each of these by the radius we get the corresponding 
ratios of the angles AOP. These ratios are very rarely used. Another 
function that is now rarely used is tlie haversino, i. c. half the versed 
sine. 

16. Projection Formulae. It is useful to remember that 
ON (i. e. the projection of the radius on the initial line) 

= r cos 6 
and iYi^(i. e. the projection ot the radius on a line perpen- 
dicular to tlie initial line) = ;• sin 0. 



THE TRIGONOMETRICAL RATIOS 11 

17. Polar co-ordinates. The position of a point P is 
determined if the distance OF from a fixed point is known 
and also the angle this distance makes with a fixed line OX 
through 0. The length is usually denoted by y and the 
angle by 6 ; these are the polar co-ordinates of P. In this 
connexion is called the pole. 

18. Graphs of Trigonometrical Functions. 

The definitions of the last suction lead to an easy method of 
drawing the graphs. On page -12 the sine graph (i.e. the graph of 
the equation «/ = sin x) is given. It is obtained as follows : 

Step 1. On the extreme left of the paper (which should be ruled 
in squares) draw a circle with its centre at the intersection of two 
lines. Take the horizontal radius CA as initial line. 

Step 2. The perpendicular B'CB gives the angles 90° and 270°. 

The diagonals through C give the angles 45°, 135°, 225°, 315°. 

By stepping off chords equal to the radius, starting from A, the 
angles 60", 120°, 240°, &c., are obtained ; and, by starting at B, 
•the angles 30°, 150°, &c., are obtained. 

Only the points P, P' , . . ., on the circumference need be obtained 
as is shown in the third quadrant ; the radii are not needed for 
drawing the sine grajih. 

Step 3. Take a point as origin, some distance along the initial 
line, and, with a convenient scale, mark off abscissae to represent 
the angles 30°, 45°, 60°, &c., and, as far as space allows, mark off 
the negative abscissae. 

Stej) 4. Through the points on the circumference draw parallels 
to the initial line to cut the corresponding ordinates. These 
points of intersection are points on the graph. 

The ordinates of this graph* are proportional to the sines; if 
we divide by the radius, the actual values of the sines are found. 

The sine graph is shown on a larger scale in Fig. X. 

19. The tangent graph. To obtain the ordinates for the 
tangent graph the radii must be produced to meet the tangent to 

* The giaph of the sine is a wavy or sinuous curve. Tlie name sine 
is therefore appropriate, although it is improbable that the originators 
of the name ever drew the graph. 



-.^ — __ _ — 


7^ — 1 — 


_. / 


? 


yt 


y 


7^ 


y- 


' t 


S 


I— * 


^ i i 


\ 


:is. 


\, 


^g 


S 


\ 


\ 
\ 


\ 


\ 


p 


J5 


~t 


/ 


-»- -/- 


y 


i 


± -7- it 




y 




I 


o 


r <5> 


\^ ' 




v 


. ^ 


- -- ^o 


^ 


N 


^ 


\ 


:;:: "::""\ 


-— o ._. ,^ 


0> 4 


7 


---- - 


^ r- - -. - ^ ^ 


^^ Ij;-- <-v:'^ ^^ 


^^^ %-\ / A 7 


, ;2;..^_^^_._/7 r< 


t..:-s\//.i:.\ 


ffl_-_ ^^^^ 5, 


ffl ^'lo ffi 


^- V ^' / . :'7 " 


\: 7- - ,^r / 


-^ . i Z-_. A ^i 


--- - ±.-^^' _ ... 



- - -- ... ^.-. 


o^ 


fl> 


1 1^ ; t 


i 1 i 




; V ■- ! ^ 


\ i "" 


i\i i 


t4^ -^ i ^ 




1 1 \ 1 . o 


1 M . •*! 


' ' \ i ! 


i ^1 i 


( , , |v^ , 1 ■ ; : . j . 


! j \ 1 


\i ^ 


i '^ ill 


It- ^v^ mitx 


-1 — ^ 1 — ^ — r 


\ ' o 


1 ■ \ ' 1 ' , fn, 


! ; \ 1 


i 1 


1 \ 


i \ in 


1 It it ^ -X-- 


"^ 1 


1 : \ ' 


\ ^ 


a> do ri- <© i> ^- fo fu_\ 


... . • . . . ; i 1 . . o 




1 
















^^— -" I < 


^•^ ! 


.^ 


y^ 


/ 


y 


/ 


"^ X 7 it 


' / ' 


( 




■ 1 / i 


'/i 1 


/ 1 1 


/ ' 


/ i ■ 


' 1 1 


' -l-i- 


\ ! 


I 1 




iffl o 



44 THE TRIGONOMETRICAL RATIOS 

the circle drawn at A. This is done in Fig. TX, and a few points of 
the graph are marked, but the graph is not drawn. 

The cosine graph. Since the cosine of x is the sine of the 
complement of x, the student should be able to modify the method 
for the sine graph so as to obtain the cosine graph. 

The secant graph. This is obtained b}^ marking off along the 
respective ordinates the corresponding values of CT (see Fig. IX). 

Examples II a. 

(Answers should be given correct to 2 significant figures.) 
By drawing to scale find the trigonometrical ratios of the 
following angles : 

1. 30°. 2. 49°. 3. 79°. 4. 100°. 

5. 78°. 6. 170°. 7. 250°. 8. 25°. 

9. 300°. 10. 156°. 11. -80°. 12. 415°. 

Find the ratios of the angle ^40/' when the coordinates of P are 
13.(4,3). 14.(4,-3). 15. (-4, -3). 16. (-4, 3). 

17. (3,2). 18. (-7, -3). 19. (-5,4). 20. (63, -16). 

The following graphs should be drawn carefully and kept for use : 
21-26. A graph on a large scale for each function, for angles 
from 0° to 90°. 

27-32. A graph on a smaller scale for each function for angles 
from -360° to +360°. 

33. The blanks in the following table are to be filled with the 
sign ( + or — ) of the respective ratios : 

Angle 0"-90° 90° -180° 180° -270° 270" -360° 

sine 

cosine 

tangent 

34. If the gradient of a hill, inclined at A° to the horizon, is 
known, what trigonometrical ratio of the angle is known ? 

35. Construct an angle whose (i) tangent is 1*45, (ii) sine is '75, 
(iiij cotangent is 1*45, (iv) secant is 2*7, (v) cosecant is 2"7, 
(vi) cosine is *75. Measure each angle in degrees. 

20. Powers of the Trigonometrical Fimctions. The 
sc^uare of sin^ is written sin^^l ; and a similar notation is 
used for other powers and ratios ; thus, in general, 
sin" ^4 means (sin ^4)". 



THE TRIGONOMETRICAL RATIOS 



45 



Inverse notation. There is one exception to the above 
statement. Suppose sin A — a. then A is an angle whose 
sine is a. This is written A = sin ' ffv Similar!}^, tan"'«, 
means an angle whose tangent is a ; and so for the other 
ratios. 

If w^e wish to express -^ — - as a power of sin J., we must 
write (sin A)~'^. 



sin A 



Note. Continental mathematicians denote tlie angle whose sin is x 
by arc sin oc. Tlxis notation sometimes occurs in English books. 

Example. Determine, hy draivlng, the angle siir'^ 5. 

Step 1. Draw axes OA, OB. [ 

Step 2. Draw circle centre 0, radius 3 units. 

Step 3. Along OB mark off 07v equal 2 units. 

Step 4. Through K draw n parallel to OA cutting circle at 
FOP. Join OF, OP'. 

We now have two angles AOP, AOP' each of which has its sine 
equal to |. AOPifir.^AOP' 139°. 




Fig. XL 



46 THE TRIGONOMETRICAL RATIOS 

It is always under.stood that siii^^rt means the smallest 
positive angle that has the sine equal to a ; and similarly 
for the other ratios. 

Examples II. 

1. Find, by drawing to scale, the sine, cosine, and tangent of 30°, 
45", 60°. Verify the results by calcidation. 

2. The sine of an acute angle is ^-* ; find the cosine, tangent, and 
secant. 

3. The sine of an angle, not acute, is f ; find the cosine and 
tangent. 

4. The cosine of an angle is j%; find the sine and tangent. 

5. Draw as many angles as possible having the tangent equal 
to -8. 

6. Given that sin 63'= '89 find cos 63^ and cos 27°. 

7. Find the value of sinM + cosM, it being known that 

sin A = -3907, cos A = -9205. 
Also find the values of tan A and sec A. 

8. Given tan d = ^fi, find cot 6 and sec 6. 

9. Draw and measure an angle A such that (i) sin ^ = —'5, 
(ii) cos J. = -"5. (iii) tan.i = -'5, (iv) sec ^ = - *5. 

10. Find the value of sec- - tun^ 6 , when sec^ = r22l and 
tan 6 ■= '7002. Justify the answer by geometrical reasoning. 

11. Are any of the following data inconsistent or impossible? 
Give reasons for your answers. 

(rt) sin A = I; [b] sec ^ = f ; 

(c) sin A = i, cos A = j ; {d} sin A = •4, cos^l = '6 ; 
(e) sin .4 = '6, cos A = 'S, tan A = '9 ; 
(/)sec^l = "35, tanyl = 1*35; 
ig) tan A — 1 ; [h) sin ^1 = 1; (/) cosec .4=1. 

12. Prove, by means of the definitions in § 12, that 

cosyl = sin (90-^) and tan (^77 -0) = cot ^. 

13. Find, by drawing to scale, (a) sin 36^ and sin 144' ; (h) cos 42° 
and cos 138° ; (c) cos 246° and cos 66 . 

14. By means of graphs (or otherwise) test the following state- 
ments : («) sin (180-^) = sin yl ; (h) cos (180 + ^) = -cos^ ; 
(f) sin (90 + ^) = sin^. 

15. By means of graphs find values for sin-^ *6, tan~^ 2"5 
C0S~^ '34, C08~^ I'b. 



THE TRIGONOMETRICAL RATIOS 47 

IG. Given sin36'' = -587;^, find cos 54", sin 144', sin 21G^ sin 324'^ 

17. Given cos 53' = -6018, find sin 37% cos 127 , cos 233", cos 413^, 
cos 307". 

18. Prove that sin 117" = cos 21\ 

19. Is it possible to find angles to satisfy the following equations ? 
Give reasons. 

(i) tan^=l; (ii) cos ^ = :|^g ; 

(iii) sin ^ + cos ^ = 1 ; (iv) sin^ d + cos^ 6=1; 

(v) sec ^ = 31416; (vi) cosec ^ = i ; 

(vii) sin ^ = ; (viii) tan ^ = 100 ; 

(ix) cos <9 = 1 ; (x) sec <9 = 78. 

20. Show that (i) sine and cosine cannot be numerically greater 
than 1 ; (ii) tangent and cotangent may be either greater or less 
than 1 ; (iii) secant and cosecant cannot be numerically less than 1. 
Why is the word numerically inserted ? 

21. Find all the trigonometrical functions of 0"^ and 90\ 

22. (i) Show that the straight line whose equation is )/ — mx 
makes an angle tan"^ m with the axis of x. 

(ii) What is the tangent of the angle made with tlie axis of x 
by the stmight line joining the two points {x^, yj and {x^, y^) ? 
(iii) Show that the equation of the line joining the two points 
, - . ?/ — Vi X- —x. 

(ivj If the equation of a straight line is y = mx + c, give the 
geometrical meanings of m and c. 

23. Show that, if x be any numerical quantity, positive or 
negative, an angle can be found whose tangent is equal to x. 

Show what limitations in value, if any, exist in the case of each 
of the other trigonometrical ratios. 

24. State concisely the changes in the sign and magnitude of 
sin ^ as ^ increases from 0" to 360^. 

25. Define the cosine of an angle of any magnitude, explaining 
the conventions regarding the signs of the lines referred to in your 
definitions. Draw the graph of cos 6 from ^ = to ^ = .^ tt. 

26. Define the sine of an angle and find by geometrical reason- 
ing the values of sin 45°, sin 90^ sin 135=. 



48 THE TRIGONOMETEICAL RATIOS 

27. Define the tangent and the versed sine of an angle ; and 
find the greatest and least values which each can have. 

28. With ruler and compasses construct an angle whose cosine 
is ^; also an angle whose cosine is — ^. Calculate the sine of the 
latter angle to three places of decimals. 

29. ABC is a triangle in which AN is the perpendicular from 
A to BC. U AB = 2'9 inches, AC = 2-5 inches, AN = 2 inches, 
find the values of sin 5, cos C, tan 5, cosec C Calculate the 
length of ^C correct to one decimal place. 

30. If A, B, C are the angles of a triangle, express sin ^ (A + B), 
cos ?; (A + B), tan \ {A + B in terms of ratios of \ C. 



CHAPTER III 
ELEMENTARY FORMULAE 

21. Reciprocal Relations. 

By definitions, sin A = -, cosec A =^-^ ' 

. '. sin A cosec A = 1 : 

1 1 

1. e. sm A = T- . cosec A = - — - • 

cosec A sm A 

In a similar way it can be proved that 

cos A sec A = 1. &c. 

tan A cot A = 1, &c. 

22. Relations deduced from Pythagoras' Theorem 
(Prop. 13, p. 10). 

In Fig. YII, § 12, we have in all cases 

i. e. X- + 7/' = r-. 

Three sets of formulae are obtained by dividing in turn 

by r, x\ \f. 

f^ ' x^ iP" 

'^Divide by n, - + — = 1 ; 

but - = cos .1. - = sin A. 
r r 

Substitute, cos- A + sin- A = 1. 

The equivalent formulae must also be learnt, viz. : 

sin- A = 1 — cos^ J., sin^ = + Vl — cos'- A ; 

cos- ^4 = 1 — sin' J., cos^l = -j- Vl ~ sin"- -4. 

In a similar way the student should prove that 

tan^ A -f 1 = sec- A. 

and cot- A4 1 = cosec- A, 

1216 D 



50 ELEMENTARY FORMULAE 

23. Relation between sine, cosine, and tangent. 



Substitute, 



tan A = 


5; 

X 




^. 


t 


r 


— 






X 




r 


tan A — 


sin A 
cos A 


is proved that 


cot A=: 


cos A 
sin A 



24. Identities. By means of the relations proved in the 
preceding sections, any expression containing trigonometrical 
functions can be put into a number of forms. It is a useful 
exercise to prove that two expressions, apparently different, 
are identical ; such exercises serve to fix the relations in the 
memory and lead to facility in dealing with trigonometrical 
expressions. 

Example. Prove that scc^ A + cosec- A ^ sec- A coser A. 
[Express all ratios in terms of sine and cosine.] 



by § 21 



Bthod I. 




L.H.S. 


_ 1 1 

cos2 A sin2 A 




sin' A + cos'^ A 




%\\\^ A cos^J 




1 




sin'^ A cos' A 




1 1 




cosM ' sin'^l 



using Formula of J^ 22 



= scc-yi co?L'C-yl. l>v § 21 

Q.KD. 



ELEMENTARY FORMULAE 51 



Method 


II. 
















sec^ A + cosec^ 


•A 
















= 


1 

cos^ 


A 


+ 


1 


A 


sin^ A + COS' 


'■A 




sin'^ 


siii'^ A cos^ 


A 












1 




1 





sec^A cose c^ A = 



sin^ A cos^ A ' 
1 



cos'^ A sin^ A sin^ A cos'^ A ' 
.•. sec^^ + cosec^^ = sec^^ cosec^^. Q.E.D. 

Method III. This method is clumsy, and should be used 
only if Methods I and II have been tried unsuccessfully. 

sec'^ J. + cosec'^^ = sec^^ cosec*^, 

j^ _1_ ^ 1 _ ^ 1 

cos'^-4 sin^^ cos^^ sin'^^ ' 

i.e. if sin^^ + cos'^J. = 1. 
But sin'^^ + cos'^^ does equal 1 ; 

sec"^4-cosec-^ = 860^^-4 cosec^^. , 

Note. The introductory * if, or some similar conjunction, is vital 
to the logical statement of the work and must not be omitted. 

25. Elimination. If two equations are satisfied by the 
same value of a single variable, there must be a relation 
connecting the constants of the equations ; this is also the 
case when n equations are satisfied by the same values of 
n—1 variables. In order to find this relation we eliminate 
the variable or variables. 

Example. Eliminate 6 from the equations si7i d = a, tan 6 = h. 

By formulae tan^(9= — sTi 

sin2^ 



Substitute h^ 



1 1 



l-8in2<9 

a' 
l~a' ' 



I.e. - - — = 1. 

a^ Tt' 

The result is called the diminant of the original equations. 

D 2 



52 ELEMENTARY FORMULAE 

Examples III a. 

1 If sin A = ^§ use formulae to find the remaining ratios. 
Draw a figure to explain why some of the ratios may be either 
positive or negative. 

2. Given that tan 6 = V find cot 6 and sin 6. 

3. Find sec S in the following cases : 

(i) cos 6 = "7921 ; (iij tan d = 1'352 ; (iiij cosec 6 = 2'583. 

4. Show how all the ratios may be found when (i) the cosine, 
(ii) the tangent is known. 

5. Prove the following identities : 

(i) sin A cot A + cos A tan A = sin A + cos A : 
(ii) tan A + cot ^ = sec ^ cosec A ; 

(iii) sin 8 tan ^ + cos ^ cot ^ = sec ^ + cosec ^ - sin ^ — cos ; 
(iv) sec2 6 - cosec^ 6 = tan^ S - cot^ ; 
(v) l-2sinM = 2cosM-l; 
. . tan a + tan (:i _ siiux cos /^ + cos a si n 3 , 
'■^^^ 1 — tan a tan (:i ~~ cos a cos /3 — sin a sin j3 ' 
(vii) (1 -tanM) -r (1 ^tan^^) = cosM -sin^4 ; 
(viii) (sin A + cos A)"^ = 1 + 2 sin ^ cos A ; 
(ix) sin^^-cos^^ = (sin ^ — cos^) (1 + sin A cob A) ; 
(x) sin A cot A cosec A + cos A tan A sec A = sec A cosec u4. ; 
(xi) tan X -tan Y — (sin X cos Y—cobX sin Y) -^ cos JV cos Y; 
(xii) (tan J[ - tan B) -r (cot A-cot B) = - tan ^ tan 5 ; 
(xiii) cos^ 3 - sin* 6 = 2 cos^ ^ - 1 ; 
(xiv) (3-4 sinM) ^ cosM = 3-tanM. 

6. Prove that versin ^ = 1 -cos ^, coversin ^ = 1 —sin A. 

7. Show that the numerical value of sin^^ -^ (1 — cos^) di- 
minishes from 2 to as A increases from 0° to 180°, and illustrate 
your answer by a diagram. 

8. Which is greater, the acute angle whose cotangent is 4, or 
the acute angle whose cosecant is | ? 

9. Prove that, if 6 is an angle less than 180" for which 
1 4 sin 6 = 7c cos 6, then cos 6 ~ 2k -i- (1 + l^) : and express tan in 
terms of k. 

10. Eliminate 9 from the following : 
(i) sin d ■■= a, cos 6 ~ h; 

(ii) sin d-\ cos 6 = a, sin 8 - cos 6 — h ; 
(iii) sec 6 - tan B = a, sec 9 + tan = },; 



ELEMENTARY FORMULAE 



53 



(iv) a sin d + h cosS = p, a sin d-b C08 d = q; 
(V) a sin 6 + b cos 6 = p, «' sin 6 + 1/ cos 6 = j^'. 

11. If rf (1— sin^) = h cos 6, prove that & (1 + sin ^j = « cos ^. 

12. If a (sec ^+ 1 j = b tan 6, prove that 6 (sec d-l) = a tan ^. 

13. If a; = a cos ^ cos (f), y = a cos ^ sin ^, z = a sin ^, eliminate 
^ and 0. 

26. Ratios of complementary angles. 

Let XOF^ A° I Fig, XII) and XOQ - 90-^° ; make OQ = OP, 
and let fall QK, PN perpendicular to OX. 




K NX 



Fig. XII. 

Then the triangles QOK, POX are congruent (Prop. 8c); so 
that KQ = OX and OK = XP. 

Hence sin XOQ = ^ 

~ OP 

= cos XOP. 
i. e. sin (90 — A) = cos A. 
In a similar way it is proved that '^ 
cos (90 -A) = 
tan (90 -A) = 
Compare these results with § 11. 

What are the corresponding results when angles nro measured 
in radians ? 

27. Ratios of supplementary angles. 

Mako XOP = .1^, and XOQ = 180 ->!'', so that QOK = A°. 

* The student is expected to complete thee formulae. 



54 



ELEMENTAKY FORMULAE 



Make OQ = OP and let fall the perpendiculars PiS', QK. 

Then the triangles QOK, PON are congruent (Prop. 8 c) so that 
0K= Oy (in magnitude) and KQ = NP. But OK and ON are of 
opposite sign. 

Y 



Q 


^\ 








p 




\ 


V ) 


N 




y 








X 


\ 


y 










1 




y^ 




Y 


^ 


r~ 




< o 


¥- 


W X 



Fig. XIII. 



Hence cos,XO^ ■=-- 



OQ 

- z9^ 

OP 
= - cos XOP, 
i.e. cos (180- A) = -cos A. 

In a similar way it is proved that 
sin (180 -A) = 
tan (180 -A) - 



[ OK ) * (where [OK) denotes the magnitude 
of OK with the proper sign prefixed) 



28. Ratios of negative angles. 

Make XOP = + ^" and XOQ = - A". 

Then XOP = XOQ in magnitude. 

Make OQ = OP. 

Join PQ cutting OX at N. 

Then in the triangles PON, QON 

OP = OQ, 

ON is common, 

included angle NOI' = included angle NOQ. 
Nl' = NQ in magnitude, 
and ONP= ONQ, so that PQ is perpendicular to OX. 



* In writing it is usual to use tlu; symbnl OK to denote length 
preceded by correct sign ; it is more convenient to print {OK). 



ELEMENTARY FORMULAE 

{OX} 



DO 



Hence cos XOQ 

e. cos (-A) = cos A. 
Y 



OQ 
_ (ON) 

OP 
= cos A' OP, 



0\ 



Q 



Fic^. XIV. 



In a similar way it is proved that 
sin ( — A) = 
tan ( — A) = 
The student should also work out the ratio of 90 + ^4, 180 + ^, 
270-^, &c. 

29. By means of the hist three sections the ratios of any 
angle can be expressed in terms of the ratios of an acute 
angle not greater than 45°. For example 
cos 139' = cos (180" - 41' j = - cos 4r, 
cos 246' = cos(-114'j 
= cos(114='j 
-cos(180'-6G") 
= -cos 66° 
= -cos (90' -24^) 
= -sin24^ 



56 ELEMENTARY FORMULAE 

It is usually easy to work directly from the figure; thus in 



cos 246° = 



Fig. XV, where XOP = 246° and XOQ = 66°, 

(ON) 
OP 

OQ 
— cos 66° 
-sin 24°. 



Y 

N 


1 


Q 

K 


P 


0/ 




X 



Fig. XY. 

30. Ratios of 0° and 90°. 

If XOV =0,7' and A coincide ; so thnt XP = 0, OX = OP. 

Hence sin 0° = ^ - 0. 

If XOP ^ 90 , then PX fulls along the y axis and X coincides 
with the origin 0. In this case NP= OP and OX = 0. 



Hence 



ELEMENTARY FORMULAE 
NP 



57 



sin 90 "" = 



cos 90' 



OP 
ON 

6F 



1, 



^ „ NP NP 
tan 90^=^^= -^ = 00 



31. Ratios of 30°, 45°, 60°. 
Make XOP equal to 30", Fig. XVI. 
Let fall PX perpendicular to OX. 

Make XOQ equal to 30"" in magnitude, and produce PA" to meet 
OQ in Q. 




Fio. XVI. 



Then, by Prop. Sc, the triangles PON, QOX are congruent. It 
follows that the triangle OPQ is equilateral. 

* The syml>nl x means 'infinity', i.e. a nnml;er greater than any 
number we can imagine. 

Consider the value of 1/x as x gets smaller and smaller, 

1=10. 4 = 1000, .^j -_. 1000000, 

As X diminishes, 1/x increases, and, by making x suificiently small, 
We can make 1/x exceed any assigned value however great. This is 
expressed thus: when x = 0, 1/x = co . Or more generally, if a is a 
constant, then a/x = oo when x = 0. 



58 



ELEMENTARY FORMULAE 



Hence 

also 

0^ 

Hence sin 30*^ 

cos 30° = ' 
tan 30° = 

Similarly sin 60° = 



OP. 



PX = IPQ since PX = QN 
= \0P since PQ = OP ; 
0.Y-' = OP'-PN^ = OP'- 1 OP'- = ^^OP"". 
VB 
2 
NP 

OP 2 
Ni' 1 

f = -866, 



OP 
ON 



•866, 



tan 60" = -v/3 = 1*732. 
As an exercise the stndent should find the values of the ratios 
of45^ 

sin 45^ = 
cos 45° = 
tan 45"" = 




Fig. XVIL 

32. The very small angle. 

In Fig. XVll let the circular measure ol" the angle AOP be $. 



ELEMENTARY FORMULAE 



59 



Then arc AP - rO, NP - r sin 0, AT = r tan ^. 
Hence Area of triangle AOP =^ lOA . NP 

= |7-2sin^; 
Ai-ea of sector AOP = \r^6; 
Area of triangle^ or =\OA.AT 

But, if ^OP is an acute angle, 
Triangle AOP < sector ^ OP < Triangle AOT, 



i e. Ir^sin^ <\r'e 
i. e. sin S <6 

This relation is true for 
throughout by ;• we have 

rBin^< rS < rtan^, 



< * ;^ tan 6, 

< tan d. 
any acute angle 



(Prop. 16) 

(§7) 

(Prop. 16) 



if we multiply 



NP < arc AP< AT. 




Fi-. XVIII. 



But, as the angle diminishes, these three lengths more and 
more nearly coincide ; and are practically indistinguishable when 
the angle is very small. This is shown in Fig. XVIII, which also 
shows that ON is indistinguishable from OA. 

Hence, when 6 is very small, there is very slight error in 
saying NP = arc AP = AT, and OiY = OA. 



60 ELEMENTARY FORMULAE 

Substituting the trigonometrical values for the lengths of these 
lines, we have 

sin S = 6 = tan 6 and cos ^ = 1 , 

when 6, the circular measure of the angle, is small. 

This may also be expressed thus : The limit of or of ^^^ > 

when 6 is zero, is 1 ; or in symbols 

T sin ^ = . «i T tan <9 , ^ 
L —X- 1 an Jj -.- = 1.* 

33. Error involved. Whatever be the value of ^, it has been 
shown that cos^^ + sin'^^ = 1. 

Using the above approximations, we have 

This last statement is true only when 6"^ is so small as to be 
negligible. Hence 

If ^ is so small that $- maybe neglected, we may say that 
sin S = e, cos ^ = 1, tan 6 = 0. 

It is shown in Higher Trigonometr}'- that sin i^' = ^ gives correct 
results if }r6^ is negligible. 

Example. // accuracy /.s required to four decimal places, find 
the sine of 1 degree. 

1° = ToTj 77 radians = -01745 radian 

■017452= -000295. (We are not, therefore, justified in saying 

cos r = 1.) 
•01745' = '000005. (This does not aft'ect the first four places so 

we may use the approximation sin 6 = 6.) 
Hence sin 1° = '0175, correct to four decimal places. 



Examples III b. 

1. Write down the sine, cosine, and tangent of 
(i) 150°, 240°, 330°, 840^ 

(ii) 60°, 800°, 135°, 225"^; 
(iii) 180°, 270-^, 405°, 210°. 

2. Find the secant and cosecant of G0°, 45°, 120°, 225^ 

3. Use Ihe definitions of § 14 to find the ratios of 180-^1. 

* Fur explanation of the wonl ' limit " seo School Algtlra, I'art ii, p. 440. 



ELEMENTARY FORMULAE 61 

4. Correct, if necessary, the following statements : 

Bin (180 -.4) = cos^ ; cos (270+ ^) = -cos^ ; 
tan(180 + ^)=tan^; sec (90-^) = sec ^ ; 
cot (90 + ^) = cot ^. 

5. In a right-angled triangle the hypotenuse is 5 feet long and 
one of the angles is 60° ; find the lengths of the other two sides. 

6. A ladder 25 feet long is leaning against a wall and is in- 
clined 45° to the horizontal ; how far up the wall does it reach ? 

7. Find sin 1', correct to 3 significant figures. 

8. Find sin 10', cos 10', tan 10' correct to 5 decimal places. 

9. What angle does a halfpenny (diameter 1 inch) subtend at 
the eye when at a distance of 10 feet ? 

10. A p3st 25 feet high subtends an angle of 30' at a certain 
point on the ground. How far from the post is the point ? 

11. Find approximately the distance of a tower which is 51 feet 
high and subtends at the eye an angle Sfy'. 

12. Prove that 

tan^ 60° - 2 tan^ 45^ - cot- 80° -2 sin- 80° - f cosec- 45°. 

13. Find approximately the number of minutes denoting the 
inclination to the horizon of an incline which rises 5^ feet in 
420 yards. 

14. In any triangle show that 

cos(^ -\~B)=- cos C, sin (B + C) = sin J, tan {B +C)= - tan A . 
Write down the other similar relations. 



Oral Examples. 

Fill in the right-hand sides of the following equalities: 

1. (i) sin2^= 2. (ij sec^J-tan^^ = 

(ii) sin 45' = (ii) cos 60' = 

(iii) cos 135° = (iii) tan d = 

(iv) tan^TT = (iv) sin (180 -^» = 

(v) sinyl cot^ (v) sec (90-5) = 

3. (i; cos2 60° + sin^60" = 4. (i) cos (9 tan ^ = 

(ii) cosec'^ C = (ii) 1 - sin' x = 

(iii) cot^7r= (iii) tan 210' =- 

(iv) cos (180-^) = (iv) cos-^A = 

I v) cot^ d = i v) cos'^ i TT + sin'^ i T 



62 ELEMENTARY FORMULAE 

5. (i) 1 + cotM = 6. (i) cos-i ^ = 

(ii) sin ^ cot ^ = (ii) cos 225° = 

(iii) sin(180 + ^j = (iii) cos (90 + A) = 

(iv) cos^ 63° + sin^ 63 = (iv) cos(-^) = 

(v) tan 330° = (v) cos (180 -Z^j = 

7. (i) tanirr - 8. (i) sin (360-^) - 

(ii) sec 60' - (ii) sin-i 2 = 

(iii) sin^ J^ + cosH^ = (iii) sec^ i tt — tan^ ^ rr = 

(iv) tan (|7r + ^) = (iv) cos 0° = 

(v) tan 135° = (v) cosec 120° = 

9. (i) tan 150° = 10. (i) sec 150° = 

(ii) cos ^/ sin 6 = (ii) cos (360°-^) = 

(iii ) cos 90° = (iii) cos-^ ^Z = 

(iv) sec 240° = (iv) sin 77° cot 77" = 

(v) cot (180-^)= (v)tant7r = 

11. (i) tan 1200°= 12. (i) 008^23° + cos' 

(ii) tan (180° + ^)= (ii) cos (270° + 5) 

(iii) tan|7r= (iii) sin-^ •4 + cos^(^= 

(iv) tan 15° cot 15° = . (iv) sin.(-(/)) = 

(v) tan-M-l)= (v) sin 225° = 



Examples III. 

1. Prove, from first principles, that sin (90 + ^) = cos ^, 
cos(180 + ^) = -cos^, tan(360-^) = -tan^. 

2. Show that sin (180-^) = sin ^, when A is (i) obtuse, (ii) 
between 180° and 270°, (iii) between 270° and 360°. 

3. Show that cos (90 - ^) = sin J., when A is (i) obtuse, (ii) 
between 180° and 270°, (iii) between 270° and 360°. 

4. Show that tan (180 + ^) = tan ^, when A is (i) obtuse, (ii) 
between 180° and 270°, (iii) between 270°, and 360°. 

5. Give 6 different solutions of each of the following equations : 

(i) sin A=\\ (ii) sin A= \. \ (iii) cos Q = ^^ ; 

(iv) tan ^ = 1 ; (v) cos ^ = - ^ ; (vi) sin ^ = - ^ • 

6. Show that all angles having the same sine as A are included 
in one or other of the forms: 180/? + ^, if n is an even integer, 




ELEMENTARY FORMULAE 63 

180 71- A, it' n is an odd integer ; and that these are included in 
the single form 180 « + ( — !)" A where n is any integer, positive 
or negative. 

7. Show that all angles having the same cosine as A are in- 
cluded in the form 360 n ± A, where n is any integer. 

8. Show that all angles having the same tangent as A are 
included in the form 180 n± A, where n is any integer. 

9. What do the forms of the three previous examples become 
when the angle is measured in radians ? 

10. If a small angle equals A°, what is the value of sin A ? 

11. Show that using the approximation sin ^ = ^ is equivalent 
to regarding a circle as a polygon with a large number of 
sides. 

12. What do the following equalities become when the angle S 
is so small that 6^ is negligible ? 

(i) sin2^ = 2sin^cos^; "^' 

(ii) cos 2 ^ = 1 - 2 sin2 d ; 
(iii) sin (S + cf)) = sin 6 cos ^ + cos 6 sin 'p ; 
(iv) C = G tan ; 

(v) t'2 = 4^,.sin2,i^. 

13. Two strings are tied to two pegs A and B in the same hori- 
zontal line, and knotted together at C; when the strings are 
pulled tight, it is found that ^C is 18 inches long and that the 
angles CAB^ CBA are 30^ and 60" respectively ; how far apart are 
the pegs and how far is C from AB ? 

14. An inclined plane, length 4 feet, is inclined at 30" to the 
horizontal, what is the length of the base ? 

15. A pendulum is held so as to make an angle of 30" with the 
vertical, what is then the distance of the end of the pendulum 
from the vertical line through point of support ? 

16. Prove the following identities: 
(i) cot^ A cos^ A = cot^ A — cos'^ A ; 

(ii) sec"^ A — sm"^ A = ta.n'^ A + coi"^ A ; 
(iii) sin^ (cosec ^ — sin 6) = cos^^; 
(iv) (cos A + cosec A) (sin A + sec A) 

= 2 -f sin yl cos ^ -F sec A cosec A : 
(v) (cos J. -f sec A) (sin.l -^ cosec .^j 

= sin A cos .1 4 2 sec A cosec A : 
(vi) sec A - sin A tan A = cos A ; 
(vii) (sec A — cosec ^) (sin A + cos ^Ij 4 sec^ ^-1 cot .4 = 2 tan A. 



64 ELEMENTARY FORMULAE 

17. (i) If 6 and cf) dift'er by ^ n, prove that tan 6 tan = — 1 ; 
(ii) Show that the lines whose equations are, respectively, 

1/ = mx and y = inx, are at right angles if mm' = — 1 ; 

(iii) Show that the graphs of the equations ax + hy + c = 0, 
ax + b'y + c = are at right angles if aa+hb' = 0, and are 
parallel if a/a' = b/V. 

18. If tan 6 = b/a, find the value of a cos d + h sin 6. 

19. If tan^ B = b/a, show that «/cos 6 + b/sin 6 = a^ + &3) 2. 

20. Give a general formula for all values of A which satisfy the 
equation cos J. = — 1 . 

21. If a sin^ 6 + b cos^ 6 ^ c and a cos'^ /9 + & sin'^ (9 = d, prove that 
« + & = c + rf. 

22. From the vertex Cof an equilateral triangle ABC a perpen- 
dicular CD is let fall on AB ; DC is produced to E so that CE 
equals CA, and JLJ? is drawn. From the resulting figure find the 
sine, cosine, and tangent of 15° and 75°. 

23. A is an angle between 180° and 270°, also cos ^4 ^ -'i\ find 
the value of cosec A + tan A. 

24. Define the cosine of an angle of any magnitude and express 
the cosine of an angle between 180° and 270° in terms of each of 
the other trigonometrical ratios. 

If cos ^ = — 65? fin^^ sin^, sec B, coiB, and explain any double 
signs which occur in your answer. 

25. Prove the following identities : 

(i) (sin Jl cos J5 + cos A sin B)"^ + (cos A cos B — ^m A sin BY= 1 ; 
(ii) sin^ B + cos*^ ^ = 1 - 3 sin^ B cos^ B ; 
(iii) cot .4— tan A = sec A cosec ^ (1 — 2 sin"^) ; 
(iv) (1— sin ^-cos A'f = 2 (1-sin A) (1 —cos A) ; 
(v) (2 cos J. — sec -4)-r (cos yl-sin ^)= 1 + tan A. 
(vi) (3 sin B cos^ B - sin^ B)^ + (cos" B-Sco&B sin^ Bf = 1 ; 
(vii) sec' B - tan' ^ = 1 + 3 tan^ B sec^ B ; 
(viii) versin (270° + ^) . versin (270°-^) = cos^ J. 

26. Prove that 

cos (180° -yl) = -cos^, and cos (90° + .4) = -sin.l. 
For what values of A is tan A = v/8 and sec A = -2 ? 

27. Solve for x the equations : 
(i) .rM 2 .;>• sec o 4 1 = 0; 

(ii) x"^ + 2 X cos a = sin^ a ; 

(iii) x"^ + (tan a + cot a) a; + 1 = 0. 



ELEMEXTAKY FORMULAE 65 

28. Prove that the number of ^^ecolKl^^ in an angle whose circular 
measure is unity is 206,265. 

The moon subtends at the eye of an observer an angle of 30', 
its distance is 240,000 miles, find its radius. 

29. If tan-^ = |, find versin 0, and explain the double result. 

30. Eliminate from 

( i) a tan + bcotd = c, a' tan ^ -f /^' cot ^ = c' ; 
(ii) a tan ^ + & sin ^ = r, a tan 6 + h' sin 6 = >:'. 



Revision E:samples A. 

i. Define the tangent of an angle. From your definition find 
tan 45^ and tan 135^, and prove that tan iItt-O) = cot 6. 

2. A surveyor goes 10 chains in a direction 35^ S. of E., then 
7'8 chains U^ E. of S. ; then 5'6 chains 10' N. of W. Find by 
drawing how far he is now from his starting-point. 

3. Prove the relation 1 -sin-^ = cos- .4 for the case where A 
lies between 90 ~ and ISC. 

Show that (sin ^ + cos^j* = l -f 4 sin.4 cos .4 + 4sin^^-4sinM. 

4. The gi-adient of a railway is 1 in 270 ; find the inclination 
to the horizontal to the nearest second. 

5. When the sun's altitude is 60', find the length of the shadow 
cast by a vertical rod whose length is 10 feet. 

6. Draw the graph of cos x between x = 15" and x — 135' with- 
out usinor tables. 



7. Explain how to find the length of the arc of a circle of given 
radius, when the angle subtended at the centre is given in degrees. 

A wheel, radius 4| feet, rolls along the gi-ound ; what hori- 
zontal distance does the centre travel when the wheel turns 
through 157' ? 

8. Why is the secant so called ? Prove that the secant is the 
reciprocal of the cosine. 

Given sec A = 2^, find tan A and sin A. 

9. Show that the graph of the straight line y = 2x-Tj is in- 
clined to the axis of x at an angle tan"^ 2. Verify this by a care- 
ful drawing. 

10. Trace the changes in sin ^ as 6 changes from 0" to 360' and 
exhibit these changes by means of a graph. 

121C E 



6Q ELEMENTARY FORMULAE 

11. Find the smallest angle which satisfies the equation 

3 cos^ + 2sin2^ = 0. 
Give also four other solutions. 

12, If sin A = f, prove that sec A + 1/cotyl = 2. 



13. What is a radian ? Prove the formula 

arc = rx 6. 
Show that if 6 is small, sin ^ = ^ api^roximately. 

14. Show that tan (180 n + A) = tan^ where n is any integer. 
If tan 3^ =^3, state three possible values for A that do not 

differ by 360°. 

15. Find the value of the expression cosec yl — j: cot yl, if 
sin A = ff (i) when A is acute, (ii) when A is obtuse. 

16. Prove the identity 2 sin ^4 cos ^ = (2 tan J) H- (1 + tan^ A). 

17. In a triangle ABC, C == 90°, AB = 15, sin ^4 = '37 ; find the 
length of^C and 5(7. 

18. Criticize the following statements: 

(«) sin2 (9=4; (h) sin 6 tan ^ = 1 ; 

(c) sin-^ ( - -3) = 170° ; {d) sin ^ + cos^ ^/sin 6 = tan 9. 



19. Explain clearly what is meant by latitude. 

A place has latitude 30° N., what is its distance from (i) the 
earth's axis, (ii) the Equator, measured along the surface ? (Radius 
of earth = 4000 miles.) 

20. Give a definition of cosine that applies to angles of any size. 
Prove that cos (180 — ^1) = — cos^. 

If sin A = l^ and A is obtuse, find cos A. 
21.. Prove that 
(cos A cos 5 + sin A sin i?/ + (sin ^ cos 5 - cos A sin Bf = 1. 

22. Draw the graph of y = seer from ;r = 0° to .r = 180°. 

23. What is meant by the statement that tan 90° = oo ? 
Is sec 90° equal to tan 90° ? Give reasons. 

24. Construct an angle A such that cos J = -5 and tan^l is 
positive. 

25. Name the points of the compass between West and South. 
How many degrees are there in the angle between SW. by S. 

and S. by E. ? 



ELEMENTARY FORMULAE 07 

26. Find the values of sin45% cts^^ J. tan \7t. 

27. Prove by means of a figure that 

sin- A + tan- A = sec- A - coa- A. 
Is this true when the angle is measured in radians ? Give 
reasons. 

28. Construct an angle such that its tangent = ^ and its versine 
is greater than unity. 

29. Prove the identity cos^ J. - sin-^l = (coL-^-1) -h cosecM. 

30. Find the value of tan from the equation 

3 tan-^' = 2-v/3tan(9-l. 
Hence find three different values of ^ that satisfy the equation. 



3L Write down six positive angles which have the same cosine 
as the angle a ; and find the positive values of 6 less than two 
right angles which satisfy the equation 
sin4^ = cos 5^, 

32. Show how to find by calculation the value of sin 30' correct 
to four decimal places. 

Verify, by substitution, 

(i) sin 60' = 2 sin 30' cos 30' ; 
lii; sin 120' -sin 60' = 2 cos 90' sin 30' ; 
liiij cos 60' - cos 120' = 2 sin 30' sin 90^ 

33. Prove the identities : 

(i) cosec- J. — cotan- .4 = 1 ; 
.. 5jfl3^nirf , 12 -13 cos 6i _ 
* "^ r2 + 13'cos e^ 5 - 13 sin ^' ~ 

34. A steamer travels along the equator from longitude o7V W. 
to longitude 5' 30' E. in 4 days. ^Yhat is the distance travelled 
in nautical miles ? What was her average rate in knots ? 

35. What is meant by the chord of an angle ? 
For which angle is the chord equal to unity ? 

Explain how to draw an angle when a table of chords is 
given. 

36. Express the following ratios as ratios of angles not greater 
than 45' : 

sin 172', cos 412', tan 246', sec 76', cosec 147', sec 236', 
cot 138=, cosine 150°, sin 67', tan 102°. 
E 2 



68 ELEMEI^TAEY FORMULAE 

87. If the circumt'ercii'es ot* the quadrants of two circles be 
divided similarly to the right angles they subtend, what would be 
the radius of a circle divided according to the French scale, in 
which the length of the arc of one grade would be equal to the 
length of the arc of one degree on a circle whose radius was 
18 feet? 

38. Point out which of the trigonometrical functions are never 
numerically less than unity, and which may be either less or 
greater than unity. 

Express the numerical values of sin 135° and tan 150° with their 
proper signs. 

39. If n be a positive v/hole number, show that the angles 

(2??. 180° + ^) and {(2;? 4 1)180°-^} 
have the same sine as y1. 

Express these in a single formula. 

40. Distinguish carefully between (sin A)~^ and sin"^^. 
Show that cos-i |. + 2 sin"' i = 120°. 

41. Trace the changes in sign and magnitude of the expression 
cos re -sin a; as .r increases from to 27r. Illustrate your answer 
by a graph. 

42. A church spire, whose height is known to be 45 feet, subtends 
an angle of 9' at the eye ; find its distance approximately. 



43. What is meant by tan"'' m ? 

If y= mx + c represents a straight line, state the geometrical 
interpretation of the coefficients m and r? 

What is the angle between the lines whose equations are 
ij = x-4:, y = a/Sx + 2'? 

44. Show that the equation of the line joining the points 

45. Find the equation of a line passing through the origin and 
(i) parallel to, (ii) perpendicular to, the line whose equation is 
y = mx + c. 

Deduce the conditions that the two lines whose equations are 
ax + hy + c = 0, a'x + h'y + / = 0, should be (i) parallel, (ii) per- 
pendicular. 



ELEMENTARY FORMULAE 69 

46. Find the eijuatioii of the line joining the origin to the point 
F whose co-ordinates are {x', y'). 

Find the equation of the line perpendicular to OP and passing 
through P. 

Hence show that the equation of the tangent to a circle at the 
point x\ y' is xx ■\- yy' = v"-, the equation of the circle being 
X- + ?/- = ;•-. 

47. If (;•, 6) are the polar co-ordinates of a point, what locus is 
represented by 

(i) r = 3, (ii) ^ = ^ tt, (iii) r cos = 5, (ivi r = 5 cos 6? 

48. If (x, y) are the Cartesian co-ordinates, (r, 6) the polar 
co-ordinates, of the same point, what relations connect them ? 

Express the equations of the previous example in Cartesian 
co-ordinates. 

Express (i) x^ + y'^ — 4:X + hy — 1, (ii) 3x'-f4// = 5 in polar co- 
ordinates. 

49. The sum of two angles is 3 radians, their difference is 
10 degrees. Find each angle in degrees, assuming that 
4377 = 135. 

50. A ring, 10 inches in diameter, is suspended from a point 
one foot above its centre by six equal strings attached to its 
circumference at equal intervals. Find the angle between two 
consecutive strings. 



CHAPTER IV 

USE OF TABLES 

34. It has been shown in the previous chapters that the 
trigonometrical ratios of any angle may be foancl roughly by 
drawing to scale or by means of graphs. By methods which are 
explained in more advanced books on Trigonometry, the ratios 
can be calculated to any required degree of accuracy. There are 
many collections of tables published, containing not only the 
actual trigonometrical ratios (the natural functions as they are 
called) but also the logarithms of these ratios. These collections 
differ slightly in their arrangement, but the following general 
remarks apply to most of them. 

35. Since any ratio of any angle is equal in magnitude to the 
same ratio of some angle less than 90°, it is necessary to tabulate 
the ratios only for angles between 0° and 90°. Thus 

sin 156° = sin (180° -24°) = sin 24°, 
cos 215° = cos (1.^0° + 35°) = - cos 35°. 
But the tables may be made even shorter, for any function of 
an angle between 45° and 90° is equal to the complementary 
function of an angle less than 45°. Thus 

sin 76° = sin (90° -14°) = cos 14°, 
^ tan69°= tan(90°-21°) = cot21°. 
This fact is used in two different ways. Some tables give all 
the ratios for angles from 0° to 45° ; so that if, for instance, 
sin 72° is required, it must be looked up as cos 18°. Other tables 
give the values of sine, tangent, and secant for angles from 0°to 90° ; 
in this case, cosine, cotangent, and cosecant must be looked for as 
the si]ic, tangent, and secant respectively of the complementary 
angle. 

The slight mental work involved is avoided by giving each 
column a " footing " as well as a heading. Thus '26892 is, in some 
tables, found on a page headed Natural Sines, on a level with 15° in 
the extreme left hanl column and under 36', i.e. '26892 = sin 15°36'. 



USE OF TABLES 71 

Bat the same page has Natural Cosines at the bottom, "26892 is on 
same level as 74° in the extreme right-hand column and above 24', 
i.e. -26892 = cos 74°24'. 

A few minutes' inspection will make the arrangement of any set 
of tables quite clear. 

36. Logarithmic Functions. Since the sine and cosine 
cannot be greater than unity, their logarithms cannot be greater 
than zero ; hence these logarithms have a negative characteristic. 
In order to avoid difficulties of printing it has been the custom to 
add 10 to all these logarithms, and to the other logarithmic func- 
tions. The values thus tabulated are called Tabular Logarithms 
and are denoted in writing by L, thus L tan 75^ = log tan 75'' + 10. 

Some of the modern tables give the ordinary logarithms with 
the negative characteristics. 

When tabular logarithms are used it is advisable to sub- 
tract 10 mentally and to work with the correct logarithm. 

37. Interpolation. It is impossible to give the ratios for all 
angles. Four-figure tables usually give values for every 6', seven- 
figure tables for every 1'. Intermediate values may, in some tables, 
be found from side columns giving the differences^ as in the case 
of ordinary logarithms. If these side columns are not given, the 
method of proportional parts * must be used. This method is 
equivalent to assuming that the graph of the tabulated function 
may be treated as a straight line for portions lying between the 
points corresponding to two consecutive tabulated values. The 
practical use is easily followed from an example or two. 

Example i. Given that 

sm28°9'.= -4717815, ami sin 28^ 10' = -4720380, 
find sm 28=^9' 43". 

sin 28^ 10' = -4720380, "000004275 x 43 

sin28'' 9' = -4717815. '00017100 

Increase for 60" = '0002565 ; 1282 



Increase for 43" = |^ x '0002565 '00018382 

= '0001838; 
sin 28" 9' 43" =-4719653. 

^ For a fuller treatment see School Algclm, Part IT, p. 376. 



72 USE OF TABLES 

In practice the zeros are omitted as in the following example. 
Example ii. Given that Zo^ cos 73° 15' = 1-4058617, and 
log cos 73^16' = 1*4053816, ^/j72cl the angle ivhen the log cosine is 
1-4056348. 

Denote the angle hy 73° 15' x'\ 

- log cos 73° 15' = T-4058617. log cos 73= 15' - 1-4058617. 
log cos 73° 15' x" = T-4056348. log cos 73° 16' - 1-4053816. 
Decrease for £c" = 2269. Decrease for 60"- 4801. 

X 2269 2269 

^^"^^ 60^4801- ^1 

.r-28. 4801)136140 

4012 
171 
.". required angle = 73° 15' 28" to the nearest second. 

Note. It is important to recollect that cosine, cotangent, 
cosecant, and their logarithms decrease as the angle in- 
creases ; consequently proportional differences must be suh- 
tr acted, not added. 

If the graphs of the functions are carefully drawn, it is seen that 
in some parts they approach much more nearly to straight lines 
than in others. It follows that the method of proportional parts 
is more accurate for some angles than for others. For a complete 
discussion of Proportional parts see Nixon's Elementary Plane 
Tngonometnj (Clarendon Press) or any advanced textbook. 

Examples IV a. 

Find, from tables, the natural function of the following angles, 
find the logarithm of the number found, and then look up the 
logarithmic function in the tables. There may be a slight dis- 
crepancy in the fourth decimal place. 



1. 


sin 17= 15'. 


2. 


cos 73° 47'. 


3. 


tan 16° 39'. 


4. 


cos 23° 19'. 


5. 


sec 67° 15'. 


G. 


cotan44°5'. 


7. 


tan 78° 53'. 


8. 


sin 83° 43'. 


9. 


cos 63° 28'. 


10. 


sin 156° 17'. 


11. 


tanl7G°16'. 


12. 


cot 100° 10'. 


13. 


cos 137° 42'. 


14. 


sin 126° 37'. 


15. 


tan 173° 14'. 



Explain carefully the difficulty that arises in connexion with 
some of the angles. 



USE OF TABLES 73 

16. Find the Cartesian co-ordinates of a point whose polar co- 
ordinates are (i) 17, 16°; (ii) 25, 114°; (iii) 49, 227^ 

Find the angles less than 180' which are determined by the 
following data : 

17. sin e = '8732. 18. cos A = _'8469. 
19. sin 5= -9340. 20. logtan .4 = 1*7932. 
21. Lcos^= 9-7432. 22. sec B = 2'5732. 
23. logsin^ = 1-3465. 24. L tan .4 = 10-4385. 

25. L cote = 10-7386. 

Find the sine, cosine, and tangent of the following angles, 
which are measured in radians : 

26. ^n. 27. iV-- 28. 1-2. 29. ^;r. 

30. Verify that 

sin 112^ = sin 70' cos 42' -f cos 70° sin 42'. ' . 

31. Find from the tables the values of ■ I 

sinjV77 and sin 27' 18'/cos 32"' 45'. " ' — • 

32. Employ the tables to verify the formula 

cot 24° 45' - cot 49= 30' = cosec 49' 30'. 

33. Find the values of cos 110", cot 160°, sin 250". 

A quantity fi is such that /n = sin //sin ;• ; complete the following 
tables : 





I 


r 


H- 


34. 


W 


12" 




35. 


26° 18' 




1-427. 


36. 




31° 52' 


1-467. 


37. 




53° 49' 


1-5. 



38. Find the polar co-ordinates of points w^hose Cartesian co- 
ordinates are (i) (3, 7) ; (iii (-3, 7) ; (iii) (-3, -7) ; (iv) (3, -7). 

39. The angle of friction e and the coefficient of friction fx are 
connected by the relation /^t = tane. Determine the missing 
quantity in the following cases : 

6 40'' 15' i 17=39' I 47' 8' : | 

,1 I -67 I -37 ' -50 

40. In a circle of radius 17 find the lengths of chords subtending 
angles (i) 37°, (ii) 73", (iii) 143° at the centre. What are the 
areas of the corresponding segments V 



74 



GEAPHS 



38. Graphs. 

Example. Draw the graph of 

3 sin (^+30°)- 2 cos (a;-30°) from x =^ 0° to x =^ 120°. 
In other words, draw the gra2)h of 

^j = S sin {x + 30°) - 2 cos (ic- 30°). 



X 



•500 


15° 


30^ 


45° 


60° 


75° 


90° 
•866 


105° 


120^ 
•500 


sin (x + 30^) . 


•707 


•866 


-966 


1-000 


•966 


.0. 


cos (x-30°) . 


•866 


•966 


1-000 


-966 


•866 


•707 


•500 


•259 


0-000 


3 sin (x + 30°) 


1^500 


2-121 


2^598 




3-000 


2^898 


2^598 


2-121 


1-500 


2 cos (.x-30°) 


1^732 


1-932 


2-000 




1-732 


1^414 


•1-000 


•518 


•000 


y 


-•232 


•189 


-598 


-966 


1-268 


1-484 


1-598 


1-603 


1-500 



The graph is shown in Fig. XIX. 

Use of Graph. 

Interpolation. The value of the function can be found for any 
intermediate value of the angle. From the graph it is seen that 
v/ = r07 when x = 50°, and y = 1'54 when x = 81°. Calculation 
shows that the correct values are 1'075 and 1*542 respectively. 

This is a useful method of testing the accuracy of a graph. 

Maximum and Minimum. When, as x increases, ij continually 
increases to a certain value and then decreases, that value is said 
to be a miaximum ; similarly, when j/ first decreases and then 
increases there is a minimum value. These maximum and 
minimum values are clearly shown on the graphs ; the corre- 
sponding points are called turning-points. 

From the graph the maximum value of 

3 sin (.r + 30°) - 2 cos {x - 30°) 
is found to be 1*62, and the correi^ponding angle is OO"". 



GRAPHS 



75 











x 










S 


1 - it 


I 1 


t ^ 




\f\ 










o 


1 ^ 


^\r -^^ 


\^ J 


JV 


^ Ul 


4^5 


I^^ 1 


X-^ 


^^di i 


v5 ° 


\^ ^ 


- ^x 




^!s 


^^ \^^ "^ 


^s: ^ ' 


T ^^ 


^ 


oK 


^, o 


\ (0 


^ 


\ 


\ 


^^ ^ 


\ - 


\. 


\ 


V 


h ib ^ 


•- -■ o n 


















-__L 



76 GRAPHS 

Kate of change of the function. The graph shows that, when 
X changes from 15° to 30°, the increase in y is more than when x 
changes from 60° to 75°; consequently the curve is steeper 
between 15° and 30° than between 60° and 75°. Thus the rate at 
which y changes compared with x is shown by the steepness of 
the curve. 

Join two points P and Q on the graph, and draw PK, QK 
parallel to the axes to meet in K. Then 

increase in v/ KP , ■ n , i i nr^r.- 

^ — i' = = tangent of the angle PQK 

increase in x QK 

= tangent of the angle PQ makes with the axis of x. 

This is called the slope of the line PQ. 

When Q approaches indefinitely near to P, the chord PQ 
becomes a tangent. Hence 

The rate of increase of y at the point P is measured by the 
slope of the tangent at the point P. 

Notice that the slope diminishes in the neighbourhood of a 
turning-point and is zero at the turning-point itself. 

39. Solution of equations. By finding where two graphs 
intersect or where one graph intersects the axis of x or a line 
pai'allel to the axis, equations can be solved just as in Algebra. 

Example. Solve the equation 

3 sm (a;-}- 30°) -2 cm- (a; -30°) - r5. 

It is seen in Fig. XIX that the graph cuts the line whose 
equation is ?/ = I'S where x = 77° and x = 120°. These are, 
therefore, the solutions within the range of the graph. 

Examples IV b. 

(The graphs should be verified in the way that the example 
of § 38 is verified.) 

Draw the graphs and find the turning-points of: 

1. sin^ X from it; == 0° to a; = 90°. 

2. cos 1 X from x= -90° to x = 90°. 

3. sin ^ ic -f cos ^J- a; from x = 15° to x — 135°. 

4. I tan {x-m°) from x = 0" to x - 90°. 

5. ^-sin e from ^ = to ^ - ^tt. 

6. sec X — tan x from O'' to 90°. 



GRAPHS 77 

7. cos- i ar-f sin- ] x from to 360\ 

8. Draw the graph of sin .r + cos.r between o-- = and x = 360°. 
Solve sin 3-+ cos .r = '89, and find the slope of the graph at the 
points corresponding to these values of ,r. 

9. Draw the graph of cos.r between the values of and 2 tt for x\ 
Show that an acute angle can be found to satisfy the equation 
X = cos X. 

10. Draw the graphs from x = -1 to j^^ = -f 1 of (i) sin-^r, 
(ii) cos-^rr, (iii) tan-^a;. How are they related to the graphs of 
sin.r, cos.r, tana? respectively? 

11. Draw the graphs whose polar equations are 

(i) r sin 6 = 11 ; (ii) r =10 sin ; 

(iii) ;• = 10 cos 6 : (iv) tan 6 = 2-45. 

12. Find from your tables the values of cos 2 x for the values 
0^ 10", 20% 30^ 40^ 50\ 60^ of .r. 

Draw the graph of cos2.r-cosa; as x increases from 0° 
to 60°. 

13. Find, by drawing graphs of 2 sin .4 and sin 2.4, for what 
values of A, less than 90', 2 sin ^ — sin 2.4 = 1, 

14. Find, by the aid of the tables, the values of sin ,r — tan2a: for 
the values 0=, 10^ 20", 45°, 60' of x. 

Make a graph to give the values of sin.r — tan2.r from ,r = to 
X = 60°. 

15. Make a table giving the values of cos^ at intervals of one- 
fifth of a radian from ^ = to ^ = two radians, taking the radian 
as 57^30'. 

From your table plot the graph of t^cos^; and hence find for 
what value of 6, between the limits and 2, ^cos^ is greatest. 

16. Plot the function ^ {sin ^-f. sin 2 (^ + 20°)} between <9 = 0^ 
and 6 = 180°, and find the maximum and minimum values of the 
function which occur within this range, and the corresponding 
values of d. 

17. Draw, in the same diagram, the graphs of sin a- and 2 cos a; 
between x = 0° and x = ISO". Show how to find from your 
diagram an angle whose tangent is 2. 

18. Taking tt as 8*1416 and using your tables, find the values of 
^-sinf? when 3 = ^7r, ^tt, ^ tt, ^tt, f^rr, and -^ tt ; and hence 
make a graph to give ^ — sin ^ from 6 = to 6 = Itt. 

19. Draw the graph of (i) sin~^a'-l-cos~^ic ; (ii) sin-'(l/a:). 



78 SOLUTTON OF EQUATIONS 

Solution of Equations. 

40. To solve a trigonometrical equation, 
(i) express all the ratios involved in terms of one ratio, 
(ii) find the value of tliis ratio by ordinary algebraical 
methods, 

(iii) find the angle from the tables, 
(iv) give the general solution. 
Example i. Solve 2 sin .r-f 3 cos a? = 2. 
Express in terms of sine, -^ 

+ 3 V*! - sin^iP =2 — 2 sin x. 
Square 9 — 9 si n^ a; = 4 — 8 sin x + 4sin^ x. 

Transpose ISsin^aj-S sina7-5 = 0. 

Factorize (13 sin x + h) (sin x-\) = 0. 

.'. sin ^ = — j5 or 1. 
Substituting in the original equation, we find that : 

(i) If sin. r = — j%, cos^r = \'i = '9281. 
Hence the bounding line is in the fourth quadrant. 
From the tables it is found that cos 22° 37' = -9231. 
Hence the smallest positive angle satisfying the equation is 

360"-22"37' = 337''2r. 

But we may add or subtract any multiple of 360° without 
altering the position of the bounding line ; hence any angle 
satisfies the equation whose value is 360° n + 337° 23', where n is 
any integer positive or negative. This is the general solution. 

(ii) If sin a; = 1, cos.^ = 0. 
The smallest solution is x = 90°. 
The general solution is 360°n + 90° or (4n + l)90°. 

Note. The same difficulty has arisen here tliat arises in Algebra 
when the original equation contains surds. After we have squared, 
the i-esulting equation is exactly the same as if we liad started with 
the equation 2 sin x — 3 cos x = 2. For this reason, after we found the 
value of sin 9, it was necessary to substitute in tlie original equation 
to find the corresponding value of cos x. 



SOLUTION OF EQUATIONS 

Example ii. Solve tan- ^ + 4 sin'^ A = 5. 
Express in terms of tan ^. 



79 



tanM + 4 



tan^^ 



1 + tan-^l 
tan^ ^ = 5. 
4 log tan ^ = -6990. 
log tan ^ = -1747. 

= log tan 56° 13'. 
^ = 56° 13'. 

Consideration of the fundamental figure shows tliat the general 
solutionis ^ = 180>i + 56° 13'. 



Multiply by l + tan^^ 
Take logarithms 

Use tables 

Hence a solution is 



41. 



General Solutions. 



General solutions can always be obtained by mentally 
considering the possible positions of the radius vector that 
give angles having the same function as some angle already 
found. This is what has been done in the two preceding 
examples. It is, however, useful to know the formulae that 
give these general solutions. 

Find an expression for all angles that have a sine 
equal to sin oc. 

We have to solve sin 6 = sin a. 




Fig. XIX a. 

The bounding line may have either of the positions shown 
in Fig. XIX a. 



en 



80 SOLUTION OF EQUATIONS 

Thus the line may revolve through n.r, where ii is even, and then 
go on CX, or may revolve through mr, where )i is odd, and th 
come back Oi. Hence ^ = n rr + 3c if n is even, 

or nrr — y if 71 is odd. 

These are included in the one formula 
e= 1177 + {-iyo(. 
If sin X = sin A, then x = ISO n 4 ( - 1 j" A. 

Exercises. In a similar way prove that 

i. 6 = 2mr±0(, when cos d = cos 0( ; 

X = S60 u ± A, when cos.r = cos^. 

ii. 0=1177 +CX, when tan^=tanCX; 

;r ='180 n + ^, when tan .-c = tan ^ . 

Example. Solve sin 3^ = cosb^. 
This is the same as 

sin 36 = sin (i/r-5^); 

3^ = n7r + (-l)"(lrr-5^). 
If n is odd Sd = mr-}>7T + 5e; 



20 = 1 



- n TV. 



Put 11 = - 2 i? + 1, then = ^; 77 - 1 rr, where ^; is any integer. 
If n is even 3 ^ = ??. tt + (In - 5 ^) ; 

Put n = 22) ^ = P^T^ + it^ ^' where p is any integer. 

The complete solution is 

d =2)rr- ^ 77 or jj Itt + ^tt. 

Examples IV c. 

Solve : 

1. 2cos2(9 = 3(l-sin^). 2. sin^ + cos^=l. 

3. sin + cos = 72. 4. 12 tan^ .4 - 13 tan A f 3 = 0. 

5. 2 008^0-- 1 = l-sin^.r. 6. sin 3^ = sin 4^. 

7. 3cot^<9-10cot2^+3 = 0. 8. 2sin .4 = tanvl. 
9. tan^ + 3cot^ = 4. 10. sinj.r + ^) = cos(a;-^). 

11. tanM + 4sinM = 6. 12. v'3tan2^ + 1 = (UV3)tan^. 

13. cosec^ = cot ^+73. 

14. cos(135° + ^)+sin(135°-^) = 0. 

15. cos^ ^ - cos ^ sin A - sin^ yl = 1. 

16. cos3^ + sin(9 = 0. 17. 3tan2 2^ = l. 



USE OF TABLES 



81 



18. tan2.r = tan2/;r. 

20. 2cos2^-f3cos^-l =0. 

22. l-7sin^--73 = 0. 

24. 2tan2^ + 7tan^-f3 = 0. 



19. 2sin2^-3sin^-2 = 0. 
21. tan2^ + sec-^-2. 
23. 3 sin (9 + 2 cos = 2. 
25. tan^-2cotr' = 17. 



42. Examples of the use of logarithms. 



Example i. Given that 
a ^ 250, h = 240. A = 72' o\ 
sin .4 



a 

sin A 



n)i B 



find B wJicn 



We have 



1. e. 



sin 5 
~V 

sin 5 



a 
b sin A 



Take logs. log sin B = log h + log sin .4 — log r/ 
= 2*3802 -2-3979 

+ r*9784 
= 2-3586 
-2-3979 
= f-9607 ; 
[.-. Lsin5= 9-9607]: 
.'. sinjB = sin66^ 
Hence jB = 180n' + (-1)» 66^ 

After a little practice the work may be arranged so that the 
logarithms are kept quite distinct from the remainder of the work. 
This same example is worked below to show the shorter method 
and the use of five- figure tables. 

Logarithms. 



sinB 
sin-B 



smA 
a"' 

240 sin 72"^ 5' ^250 
sin 65^ 59'. 



2-38021 
+ 1-97841 



2-35862 
2-39794 
1-96068 



Example 11. The sides and angles of a irianglc are con- 
nected hij the relation tan \ [A —B) — — -, cot \ C: find A a)>d 
•^ ^ ^ a-\-h 

B irhen a 

1216 



242o, 6 = 164-3, C= 54^36' 

F 



82 



USE OF TABLES 
-^-^ '^-2 cot 27=18' 





2 - 406-8 """" 




r 78-2 , , 




= tan 20° 26' 


.'. 


^? = 20-26', 


by question 


^^^ = 62-42-. 


Hence 


A = 83" 8', 




5 = 42" 16'. 



Logarithms. 

1-89321 

+ -28723 



2-18044 
2-60938 
1-57106 



The step in brackets is required if the tables do not give the 
cotangents. Since A and B are angles of a triangle, 1{A-B) 
cannot ec^ual any of the angles 180 n + 20° 26' (except when n = 0). 
so that there is no need to give the general solution. 

Example iii. // a, h, c are the sides of a triangle, 

(g y\ [g (A 

fan lA—^ ^ — -^ — —-^ iclierc s is half the sum of the 
V s(s—a) 

sides. Find A when a = 1762, h = 893, c = 1386. 

Logarithms. 



s = (1762 + 893 + 1386)-f-2 = 


4041-f-2 




= 2020-5, 




305212 


s-a= 258*5, 




280243 


s-h = 1127-5, 




+ 5-85455 


s-c = 634-5 ; 




3-30546 


^ A 71127-5x634-5. 
' * "^^^2 ~V 2020-5x258-5' 


2-41246 
-5-71792 


.-. ^ = 49° 29', 




2)0-13663 
006831 



A = 98° 58'. 



Examples I"V. 

1. Use the tables to find the values of sin 52'', cos 140^, tan 220". 
cos 340", sm340^ 

2. Divide sin52 ])y cos 52'; verify your answer liy finding the 
value of tan 52 ' from the tables. 



USE OF TABLES 83 

3. Write down by using tables the values of sin 140', cos 160", 
cos 220°, tan 320°. 

4. Find the smallest positive value of B which satisfies 

cos = sin {(4 m + 3) | tt + 0(}. 

5. Find all the values of 6 which satisfy the equation 

4cos6J-3sec^ = 2tan^. 

6. Find the inclination to the horizon of an incline which rises 
5i feet in 420 yards. 

7. Solve the equation tan-t^ - (1 + ^/S) tan (9 + ^3 = 0. 

8. Given that tsui^C = \/(s-a) {s-h)-=r s{s-c), find C when 
a = 32, b = 40, c = 66. 

9. Solve the equations cos(2x + 3 //; = |, cos(3.r + 2 //) = ^^3/2. 

10. Given log 2 = '30103 and log 3 = -47712, find (without the 
tables) L sin 60= and L tan 30° . 

11. Find the acute angle whose cosine equals its tangent. 

12. The current C in a circuit, as determined by a tangent 
galvanometer, equals G tan 6, where 6^ is a constant depending on 
the galvanometer only and 6 is the deflexion of the needle. 
Determine the ratio of two currents which give deflexions of 
27' 14', 35' 23' respectively. 

13. The length of a degree of latitude in latitude (p is 

(11 11-317 - 5-688 cos 0) 10* centimetres. 
Find the length at London (latitude 5r3rN.) and Melbourne 
(latitude 37"50'S.). 

14. The length of the seconds pendulum in centimetres, at 
a place whose latitude is A, is 99-3563 --2536 cos 2A. Find the 
length of the seconds pendulum at Paris (lat. 48° 50' N.) and 
Calcutta flat. 22' 33' N.). 

15. The acceleration of a falling body at a place whose latitude 
is X, when measured in centimetres per second per second, is 

980-6056-2-5028 cos 2 X. 
Find the acceleration at Montreal (lat. 45° 30' N . i and Cape Town 
(lat. 33° 40' S.). 

16. A quantity A is determined by the relation A = ^ ah sin C. 
Complete the following table : 





A 


a 


b 


C 


i. 




17 


43 


IT 14 


ii. 


342-6 


21-3 


38-19 




iii. 


984-2 




43-82 
f2 


43 21' 



84 USE OF TABLES 

17. Draw the graph of 

tan 6-0 from ^ i= to ^ = ^• 

Hence solve tan 6 = 6 + S. 

18. Given that A and B, the angles of a triangle, are connected 
by the relation asm B = fesin^, find B when a = 181, b = 217, 
A = 34° 15'. 

19. If 2i? = a'sin^, find the value of A when i? = 179*4 and 
a = 300. 

20. Verify that 

cos 146' 43' - cos 56" 51' = - 2 sin 44° 56' sin lOP 47'. 

21. Find the length of (i) the chord, (ii) the arc, subtending an 
angle 70° at the centre of a circle of radius 25 cm. Find also the 
area of the segment. 

22. Find the length of the side of a regular decagon (i) inscribed 
in, (iij described about, a circle of radius 2 "7 inches. 



CHAPTER V 

THE RIGHT-ANGLED TRIANGLE 

43. In the previous chapters we have had to deal with only 
one angle at a time, and have been able to draw one of the lines 
containing that angle horizontal. In applications of Trigonometry 
we often have to deal with several angles in the same example, 
and the lines containing them are drawn in various directions ; in 
such examples it would be difficult to apply the definitions of 
§ 12. But it has been shown that the ratios of any angle can 
be expressed in terms of the ratios of an acute angle. In practice, 
therefore, it will often be found advisable to use the following 
definitions, which apply only to acute angles. 

In a right-angled triangle an acute angle is contained by 
the hypotenuse and one of the other sides which is called 
the side adjacent to that angle. The remaining side is 
called the side opposite. Then in Fig. XX 




86 



THE KIGHT-ANGLED TRIANGLE 



sin BAG = 



cos BAG = 



tan BAG = 



opposite 
hypotenuse ' 

adjacent 
hypotenuse ' 
opposite ^ 
adjacent ' 



cosec BAG = 



sec BAG 



cot BAG = 



hypotenuse 
opposite ' 

hypotenuse 
adjacent ' 

adjacent 

opposite 



These are clearly the same definitions as in § 12, the triangle 
BAC taking the place of the triangle PON; and the various 
formulae proved in Chap. Ill can be proved directly from the 
definitions of this section. 

44. It is usual to denote the angles of any triangle ABC by 
the capital letters A, B, C; the lengths of the sides opposite the 
angles A, B, C are denoted by a, b, c respectively. 

Hence, in a triangle ABC, right-angled at C, 
a 



sin^ 



I.e. a = c sin A 



cos A = -, 
c 



tan^ 



&' 



i.e. b = ccos^ 



e. a = &tan^. 



Examples Va. 

1. Prove, from the definitions of § 43, that 

(i) cos ^ = sin 2? = sin (90 — ^) ; 
(ii) sin A = cos B = cos (90 — ^) ; 
(iii) tan A = cotB = cot (90 - A.). 
2. 

P 




N M 
Fig. XXI. 



THE EIGHT-ANGLED TRIANGLE 



87 



In the figure PNO, QMO, QKP are right angles. 
If 0N=o, NP=1, OM=G, MQ = o, find the values of 
sinPO.V, inn KPQ, tan KQP, sec QOM, cos KQO, cosec XRO. 

3. If, in Fig. XXI, OP = 8, POQ = 30", QON = 45°, PQO = 90^, 
find the lengths of OQ, PQ, PK, QM, OM. 

4. A circle is described on a horizontal diameter ^5 of length 
10 inches; a point C is taken on the circumference, such that 
BC = 7, and CD is let fall at right angles to AB. Find the size 
of the angle BAC and the length of CD. 

5. In a triangle, right-angled at C, a perpendicular is let fall 
from C to the hypotenuse ; prove, by Trigonometry, that this 
perpendicular is a mean proportional between the sides containing 
the ris^ht angle. 



D 


C 


B 




^\ 




A 







Fig. XXII. 



In the above figure (which is not drawn to scale) AO is at 
right angles to DE, OC is at right angles to AG, OG is at right 
angles to AO and EF', also G is the middle point of AB. 

Use this figure in the following examples. 

6. UAC= 10, CAD = 40°, find, if possible, the lengths of all the 
other lines. 

7. If CD = S, AB = 24, find sin CAD. 

8. If GF= 18, AE=b, OC = 5, find cos ACD and the length 
of^a 



88 



THE RIGHT-ANGLED TRIANGLE 



9. If AB = I, CAD = 0, find CD and AO. 

10. If CG = a, CGO = 6, find AD and AO. 




Fig. XXIII. 

With the ordinary notation for the sides and angles of a 
triangle, find in the above figure : 

11. The length of AD when c = 70, i? = 49°. 

12. The length of AD when 1= 42, C = 72°. 

13. The length of BD when c = 76, B = 39°. 

14. The length of CD when h = 114, C = 114°. 

(What geometrical fact does the negative sign in the result 
show ?) 

15. Prove that the area of the triangle = i«& sin C; give the 
proof, also, when C is obtuse. 



Solution of Right-angled Triangles. 

45. The angles and sides of a triangle are sometimes 
called the six parts of a triangle. The determination of all 
the parts, when only some of the parts are known, is called 
solving the triangle. If the triangle is known to be right- 
angled, the triangle can be solved if one side and one other 
part are known. 

Example i. A man, standinci 100 feet from the foot of 
a clmrelt stee2)le, finds that the inu/le* of elevation of the top 

* If a i)er.son is looking upwards, the angle Ins line of sight makes 
with the horizontal is the angle of elevation ; similar!}', if lie is look- 
ing downwards, the angle his line of sight makes with the horizontal 
is the angle of depression. 



THE RIGHT-ANGLED TRIANGLE 



89 



Ls 50°. If Ms cue is 5^ feet from ilie gyoiind, what is the 
height of the steeple ? 

[The figure should be drawn neatly but need not be drawn to scale.] 

In Fig. XXIY AE represents the steeple, BC the man; CD is 
drawn parallel to BA. 




[Mental. In the right-angled triangle CDE we know that 
CD = BA= 100 ft., angle DCE - 50^ 
and w"e wish to find DE. 

unknown side 



Here 



some ratio of known anglt 
= tan DCE, 



known side 
DE 
. BC 

i.e. /)i^ = 100 tan 50° feet 

= 100 xl'1918 feet 
= 119-18 feet. 
Therefore height of steeple = AD + DE= 124'68 feet. 

Example ii. The shadoir, cast by the sun on a horizontal 
plane, of a vertical pjole 10 feet high, is ohscrred to he 14 feet 
long ; find the altitude of the sun [i. e. the angle of elevation 
of the sun). 



90 



THE RIGHT-ANGLED TRIANGLE 



In Fig. XXV AB represents the pole, AC the shadow; so 
that CB is the direction of one of the sun's rays. 




loF^ 



[Mental. In the right-angled triangle BAC ^ve know 5^ and AC, 
and wish to find the angle ACB. 

ratio of known sides = some ratio of required angle.] 

— = tan ACB ; ^^^ ^^, ^^, ^ .^^^^9. 

.-. tan ACB = ]^ = -71429. 440 is diff. for 10'. 

.-. ^CJ5 = 35°32'. .-. 100 

Sun's altitude = 35° 32' to nearest minute. 



10_Q' _ 0' 

44 



Here 



sec BAC: 



Example ill. A ship C is observed at the same time from 

two coastguard stations A and B, 1459 yards apart. The 

angle ABC is found to he 90°, and the angle 

BAC to he 67° 14', ivhat is the distance of 

the ship from station A ? 

AC 
AB 

.'. log^O = logl459-+logsecG7°14' 

= 3-16406 

+ -41111 

+ 120 

= 3-57637, 
i.e. ^0=3770-3. 

Distance of ship from A = 3770 yards to 
nearest yard. 




Fig. XXVI. 



^ If tlie tables do not contain the secants, the working must be made 
to depend on the cosine. 



THE RIGHT-ANGLED TRIANGLE 



91 



AC 


1 










AB~ 


cos BAC 










log AC = 


log 1459- 


-log 


cos 


67° 


14 


= 


3-1641 










_ 


1-5877 










= 


3-5764, 










.e. AC = 


3770. 











Example iv. Ttvo men, A and B, 1370 yards axmrt, 
observe an aeropJane C at the same instant and find the 
respective angles of elevation to he 40° and 67°. If the plane 
ABC is vertical, ralcidate the height of the aeroplane. 




Fig. XXVII. 

Let h feet be height of aeroplane. 
From triangle ADC, AD = h cot 40°. 
From triangle BDC, BD = h cot 67' ; 
but AD + BD = AB; 
.-. hcoti0° + hcot6r= 137. 
1370 



cot 40° + cot 67" 
1370 



1-61622' 
.-. h = 847-62 ; 
Height of aeroplane = 848 yards to nearest yard 



cot40°= 1-19175 
cot 67°= -42447 
Logarithms 
8-13672 

- -20852 



2-92820 



Examples V b. 

1; The string of a kite is known to be 500 feet long, and it is 
observed to make an angle of 55° with the horizontal ; find the 
height of the kite. 



92 THE RIGHT-ANGLED TRIANGLE 

2. From the top of a cliff, 215 feet high, the angle of depression 
of a ship is observed to be 23° 20' ; what is the distance of the ship 
from the foot of the cliff? 

3. From a point 56 feet from the foot of a tree the angle of 
elevation of the top is 73'' ; find the height of the tree. 

4. The top of a conical tent is 9 feet above the ground ; the 
radius of the base is 5 feet ; what is the inclination of the side of 
the tent to the horizontal ? 

5. The shadow thrown by a flagstaff is found to be 55^ feet long 
when the sun's altitude is 53° 15'; what is the height of the 
flagstaff? 

6. I know that a certain tower is 144 feet high. I find that its 
elevation observed from a certain point on the same level as the 
base of the tower is 37° 16'. Find the distance of that point from 
the base of the tower. 

7. A sphere of radius 4 inches is suspended from a point A in 
a vertical wall so that it rests against the wall. The string is 
11 inches long and is in the same straight line as a radius of the 
sphere. Find the inclination of the string to the vertical. 

8. From the top of a cliff, 254 feet high, the angle of depression 
of a ship was found to be 9° 28', and that of the edge of the sea 
72° 40'; how far distant was the ship from the edge of the sea? 

9. Two observers on the same side of a balloon and in the same 
vertical plane with it, a mile apart, find its angles of elevation to 
be 15° and 65° 30' at the same moment. Find the height of the 
balloon. 

10. From the top of a tower, 108 feet high, the angles of de- 
pression of the top and bottom of a vertical column are found to 
be 30° and 60° respectively. What is the height of the column ? 

11. A flagstafF, 80 feet high, is fixed in the centre of a circular 
tower 40 feet in diameter. From a point on the same horizontal 
plane as the foot of the tower the elevations of the top of the 
flagstaff and the top of the tower are observed to be 35° and 30"^ 
respectively. Find the height of the tower, 

12. A river, the breadth of which is 2C0 feet, Hows at the foot of 
a tower, which subtends an angle 25° 10' at a point on the further 
bank exactly opposite. Find the height of the tower. 



THE RIGHT-ANGLED TRIANGLE 98 

13. A person standing at the edge of a river finds that the eleva- 
tion of the top of a to'»ver on the edge of the opposite bank is 60'' ; 
on going back 80 feet he finds the elevation to be 45'^; find the 
breadth of the river, 

14. From the top of a tower. 50 feet high, the angle of depression 
of a man, walking towards the tower, is noticed to be 30' ; a few 
moments after it was 45'. How far had the man walked between 
the two observations ? 

15. Two p6.=!ts, 400 yards apart, at the sides of a straight road 
running E. and W., are observed to bear N. 20° E. and E. 20° N. 
respectively. Find the distance of the observer from the road. 

16. Two points A and B and the foot D of a tower CD are in 
a horizontal straight line, and the angles of elevation of C, the top 
of the tower, as seen from A and B respectively, are 25" 46' and 
35=25'. If the distance AB is 200 feet, find the height of the 
tower. 

17. A vertical post casts a shadow 15 feet long when the altitude 
of the sun is 50= ; calculate the length of the shadow when the 
altitude of the sun is 32°. 

!>:. A vertical mast, having its base at A, is set up on a horizontal 
plane. B and C are points in the plane in a line with A, and such 
that the angular elevations of the top of the mast, when observed 
at these points, are respectively y and (3. If tan ^ = f , tan ,3 = f 
and the length of BC is 105 feet, find the height of the mast. 

19. A man standing on a tower at a height of 80 feet from the 
ground observes that the angles of depression of two objects on 
a straight level road running close to the foot of the tower are 60° 
and 30°. If the objects are on the same side of the tower, how far 
are they apart ? 

20. A, B, C are three points in succession on a straight level 
road, and P is another point so situated that the angles PAB, 
PBA, PCA are respectively 90°, 60°, and 45°. If a man walks at 
a uniform rate from ^ to JB in 25 seconds find, to the nearest 
second, how long it will take him, at the same rate, to walk from 
Bio C. 

21. A ray of light passes through a hole A in a graduated 
horizontal scale AB in a direction perpendicular to the scale and 
is reflected by a vertical mirror which is distant 30 inches from the 
scale and makes an angle x'' with the incident (i.e. approaching) 



94 THE RIGHT-ANGLED TRIANGLE 

ray. After reflection the ray makes the same angle with the 
mirror as before and shines on the scale at a distance 8 inches from 
A. Find the value of x. 

If the mirror now swings through an angle 1°, how far will the 
spot of light on the scale move ? 

Elementary Navigation. 




{The student sliould revise §§9 and 10 dealing with latitude and 
longitude and the points of the compass. [ 

46. When a ship is sailing, the angle between its direction of 
sailing and the meridian the ship is crossing is called the course. 

If the course is constant, the ship 
is said to sail on a rhumb-line. 
The distance between two positions 
of the ship is then measured along 
the rhumb-line. The difference of 
latitude of two places is the arc of 
a meridian intercepted between the 
parallels of latitude passing through 
the two places. The departure 
' between two meridians is the dis- 

Fig. XXVIII. tance between the two meridians 

measured along a parallel of lati- 
tude ; thus the departure between any two given meridians is not 
a constant but diminishes from the equator to the poles.* 

47. A small portion of the earth's surface may be regarded as 
a plane ; for distances small, compared with the earth's radius, we 
may therefore use the formulae of Plane Trigonometry. 

Plane Sailing is the name given to that part of navigation 
which treats the surface of the eaith as a plane. On this assump- 
tion the meridians become parallel straight lines, the rhumb-line 
becomes the hypotenuse of a right-angled triangle of which the 
departure is the side opposite to the course, and the difference of 
latitude is the side adjacent. Thus problems on Plane Sailing are 
merely examples in the solution of right-angled triangles. 

* In navigation distances are usually measured in nautical miles ; 
a nautical mile is the length of an arc of a meridian (or the equator) 
which subtends an angle of 1' at the centre of the eartli ; thus a 
distance of 75 nautical miles is usually written 75'. 



THE RIGHT-AXCxLED TRIANGLE 



95 



Examples V c. 

(The distances are given in nautical miles.) 

1. A ship sails SE. by S., a distance 81 miles; what is her de- 
parture and difference of latitude ? 

2. A ship sails N. 49"^ 41' W., a distance 73 miles ; what is the 
departure and difference of latitude ? 

3. A ship sails SSW. until its departure is 198 miles , what is the 
distance sailed and the difference of latitude? 

4. If the course is 3^ points W, of N., and the difference of 
latitude 149 miles, what is the distance ? 

5. A ship sails between North and West, maldng a difference of 
latitude 157^ miles and departure 79 miles ; what is the course ? 

6. A ship sails westward 247 miles along the equator from 
meridian 16" E. ; what is now the longitude ? 

7. A ship sails 247 miles eastward along the parallel 40" N. : 
what is the change in longitude ? 

8. When a ship sails any distance (great or small) along a 
parallel of latitude, show that 

difference of longitude in minutes = departure x secant of latitude. 

9. A ship, from latitude 54' 22' 10" N., sails 195^ miles I of a 
point S. of SE. ; what is now the latitude ? 

10. Leaving latitude 49' 37' N., longitude 15' 22' W., a ship sails 
SW. by W. 150 miles ; find the new latitude and longitude. 




Fig. XXIX. 

48. Parallel Sailing. If X' is the latitude, then the radius 
of the parallel of latitude (AT in Fig. XXIX) is cos X x radius of the 



96 THE RIGHT-ANGLED TRIANGLE 

earth. If B is the radian measure of the difference of longitude of 

two places on the same parallel, the length of the arc between 

them is 6 cos X x radius of the earth. The radius of the earth is 

21600 ,. , ., 
-^^ — nautical miles. 

a TV 

iience departure = - ?, — 6 cos X. 

When d is reduced to minutes, this relation iDecomes 

departure = difference of longitude x cosine of la,titude. 



49. Middle Latitude Sailing. In Middle Latitude Sailing, 
the departure between two places, whose latitudes are X and X', is 
taken to be the departure between their meridians, measured at 
the latitude |(X + X'). On this assumption, 

departure = diff. of longitude x cos |(A + X'). 

50. Traverse Sailing, If a ship sails on different courses, 
from A to B, from B to C, from C to Z>, &c., then, by the methods 
of Plane Sailing, the total changes in latitude and longitude can 
be worked out. This is called the method of Traverse Sailing. 
This method can only be used when the whole area traversed can be 
regarded as plane without introducing a great amount of error. 

Example. A ship left a j^osition in which Oporto Light 
(lat. 41° 9' N., long. 8° 38' T^.) lore W. hy N., 15 miles 
distant. Afterwards she sailed as under : 



Courses. 


Distances 


N.W. 


70' 


S. by W. ^W. 


55' 


E. 


35' 


N.N.W. 


42' 


S.E. 


51'. 



Find her bearing and distance from the Light in her last 
position. 

We liave a sories of right-angled triangles to solvo, the hypotenuso 
and an acute angle being given in each case. In practical navigation 
special tables are userl, calU'd Traverse Tables. 



THE RIGHT-ANGLED TRIANGLE 



97 



The angle the hypotenuse makes with the meridian is taken iu 
each case. 
Oto A. 
Hypotenuse 15', angle 7 points = 18^ 45'. ri761 

Diff. of latitude = 15' x cos 78= 45' 1-2902 

-1 2-926' S. ^63 

Departure = 15' x sin 78M5' 1-1761 

= 14-71' E. r-9916 

ri677 




C 
Fig. XXX. 

Note that bears W. by N. from A, but A bears E. by S. from 0. 
A to B. 

Hypotenuse 70', angle 45=. 
Diff. of latitude = 70 x cos 45= 

= 49-497' N. 
Departure = 49-497' E. 

Bio a 

Hypotenuse 55', angle 1| points = 16= 52|'. 
Diff. of latitude = 55 cos 16= 52 V 

= 52-64' S. 
Departure = 55 sin 16° 52i' 

= 15-97' W. 
The other triangles are worked in the same way. 



•7071 



1-7404 
1-9809 
1-7213 
r7404 
r4628 
1-2032 



98 THE RIGHT-ANGLED TRIANGLE 

Tabulate the results thus : 





Distance. 


Diff. of Latitude. 


Depai 


ture. 




N. 


S. 


E. 


W. 


E. byS 


15' 




2-93 


14-71 




NW. 


70' 


49-50 




49-50 




S.byW.iW. 


55' 




52-64 




15-97 


E. 


35' 






35 




NNW. 


42' 


S8-80 






16-07 


SE. 


51' 




36-06 




36-06 






88-30 


91-63 

88-30 


99-21 
68-10 


68-10 




3-33 


31-11 





We see now that the final difference of latitude from the light is 

3-33' S., and departure 31*11 E. ; so that we have to solve a right- 
angled triangle given the two sides. 

8*33 -5224 

31*11' 1-4929 

F0N=6°r. 1^95 

^^_ 31-11 1*4929 



In Fig. XXX tan FON = 



cos7^( 


:>N 1*9975 


= 31*29 


1*4954 


In her final position the ship bore G'^?' S. of E., 3r3 miles 


distant from the Light. 




To find thf lo7igitude of the ship. 




Latitude of 


41° 9'N. 


Diff. of latitude for F 


3*3' S. 


Latitude of F 


4V 5-7' N. 


Middle latitude 


41" 7'. 


Difference of longitude in minutes = 


departure 
cosine of middle latitude 




31*11 1*4929 


~ 


cos 41° 7' -1-8770 


= 


41*30'. 1-6159 



Longitude of i''= 8°38'-41*30' 
= 7" 57' W. 



THE RIGHT-ANGLED TRIANGLE 99 

Examples V d. 

1. Find the distance on the parallel between Cape Agulhas 
(lat. 34°50'S., long. 20M' E.) and Monte Video (lat. 34° 50' S., 
long. 56° 9' W.). 

2. A ship steamed at the rate of 12 knots from Albany 
(lat. 35° 3' S., long. 118° 2' E.) to Cape Catastrophe (lat. 35° 3' S., 
long. 135° 58' E.}. How long did she take on the voyage ? 

3. A ship sailed from Port Elizabeth (lat. 34° T S., long. 25°40' E.) 
SE. i^ S., until her departure was 397' ; find her final position. 

4. Find the course and distance from Syracuse (lat. 37° 3' N., 
long. 15° 15' E.) to Fano (lat. 39° 52' N., long. 19° 19' E.). 

5. A ship left a position from which Cape Clear (lat. 51° 26' N., 
long. 9° 29' W.) bore NE. by E. 12*5 miles distant and sailed 
South 150' and then West 290 miles. Find the bearing and 
distance of Cape Clear from the ship in her last position. 

6. Find, by Middle Latitude Sailing, the departure between 
two places whose positions are 13° S., 50° E. and 20° S., 60° E. 

7. A ship sails from 50° N., 50° W. to latitude 48° N., the 
distance being 157' ; find the new longitude. 

8. Cape Ortegal (lat. 43° 45' N., long. 7° 6' W.) bore SW. JW. 
12 miles distant. Afterwards sailed as under : 



Prue Courses. 


Distances. 


NNW.iW. 


70' 


ESE. 


85' 


NNE.fE. 


lor 


S. 


50' 


wsw. 


92' 



Find the final latitude and longitude. 

9. A ship left the Texel (latitude 52° 58' N.) and then sailed W. 
by N. 34', S. by E. 45', W. by S. 35', SSE. 44', WSW. iW. 42'. 
Find the course and distance to Dungeness which lies 139' West of 
the Texel in latitude 50° 55' N. ' 

10. A ship, latitude 17° 10' N., is making for a harbour, latitude 
13°10'N., and 180' W. of the ship. She sails SW. by W. 27', 
WSW.|W. 30', W. by S. 25', W. by N. 18', SSE. 32', SSE.f E., 27', 
S. by E. 25', S. 31', SSE. 39'. Find the course and distance to the 
harbour. 

g2 



100 



THE RIGHT-ANGLED TRIANGLE 



11. A ship left a position in which Heligoland bore ENE. 12', 
and then sailed NW. 24', S. by W. 20', NW. by W. 32', S. by E. 36', 
WNW.iW. 42', SSE. AE. 16', W.fN.45'. What is then the 
position of the ship ? Heligoland lies 54° 12' N., 7° 54' E. 
• 12. A ship sailed from Barcelona (41°25'N., 2"10'E.) SE. by 
E.|E. until she reached latitude 36°2rN. What was then her 
longitude ? 

13. A ship left a position in which Sable Island (43°24'N., 
65° 36' W.) bore NW. | W., distant 12 miles. 
Afterwards sailed as under : 

Courses. Distances. 

ESE. 72' 

SW.iW. 37' 

NNE. 42' 

E. 25' 

Required the latitude and longitude reached. 

51. The Double Angle. 

In Fig. XXXI, the Sing\eBAC = A° ; onAB [i semicircle 
is described with centre 0, so that angle BOC =2A. 
Let fall CN perpendicular to AB. 




N B 



COS 2 ^ = 



ON 

OC 

_ AN- AO 
~ OC ~ 

_AN__ 
"" OC 



m 

:hej 

— 1 hypotenuse of the triangle of ' 

2 OC which AN is a side. J 



THE RIGHT-ANGLED TRTANC^I^ 

n A ^J Fill in the vacant places with the 



_ AN AG__ 

~ ag'ab 

= 2cosM- 1. 



Exercises. In a similar way prove 
i. sin 2 ^ = 2 sin A cos A. 
ii. 003 2.4= l-2sin2^. 

Deduce 
iii. cos 2 ^ = cos^ A - sin^ A. 

. ^ ^ , 2tan^ 
IV. tan 2A = 



l-tan^^ 

V. sin ^ = 2 sin | ^ cos ^ ^ ; cos A = cos^ \A- sin^ \ A. 
vi. 2 cos*^ i ^ = 1 + cos A, 
vii. 2sin^^yl = 1-cos^. 



sin J. 1— cos^ 

viii. ■ ■ 



... , , , /I— COS^ SI 

111. tcin IA= . I, = ^ — 

'NJl + cos^ 1 + 



cos>4 sin J. 



ix. Prove the formulae for sin 2^ and cos 2^ when 2yi is 
obtuse. 

X. Do these proofs apply to angles of any size? If not, 
between what limits do they apply ? Why is the ambiguous sign 
omitted in viii ? 



52. Geometrical questions may often be solved by using 
Trigonometry. For example : 

If from a point outside a circle a secant and a tangent be drawn, the 
rectangle contained by the ivhole secant and the part outside the circle 
is equal to the square on the tangent. 



102 I'HE EIGHT-ANGLED TRIANGLE 

In Fig. XXXII it is required to prove that rect. PA . PB = eq. 
on PT. 



r> 




Fig. XXXII. 

Let radius = r, OP = c, and angle OPB = ^, angle OAC = (/). 
PA = PC -AC 

= ccos6 — rcos(j). 
PB = PC+CB 

= PC+AC (Prop. 21) 
= ccos^ + rcos0 
PA.PB = c' cos- e-7^ COS^ 

= c^ — i^ — c^ sin'^ 6 + 7-^ sin^ (p. 
But c sin ^ = OC from triangle OPC 

= rsincp from triangle OAC. 
Hence PA . PB = c" - r"" 

= OP^-OT^ 

= PT2 since OTP is a right angle. 

53. Known results in Geometry are useful for proving 
Trigonometrical relations. 

Show that, in any triangle, 

tan^jA-B) _a-h 
tan^{A+B) ~ r7T7>* 

With centre C and radius CA (i.e. ?>), describe a circle cutting 
CB in E and CB produced in D. 



THE RIGHT-ANGLED TRIANGLE 



103 



Then BE = a-h, and BD= a + h. 

Join AD and AE. 

Through E draw i?i^ parallel to DA and meeting AB at F. 

Then the angle DCA at the centre = 180 - C = ^ + 5. 

So that the angle DEA at the circumference = \{A^-B). 




Fig. XXXIII. 

Also the angle BAE = BAC-EAC= A-\{A+B) ==\{A-B). 
Also the angle EAD, being in a semicircle, is a right angle. 

tani(^ + £) = ^, 



tan ^ (^ -P) = — - , since AEF= EAD = a right angle. 
AE 



Hence 



t an^(^-^) ^^i?^ 
tan|(J. + J5) ^Z) 



= ^— since -&i^is parallel to AD. (Prop. 12 «. 

a-b 

~ a + b' 



Corollary. ^ + B+C=180^ .-. ^(A + B) = 90-^C. 
Hence the above result may be written 

tan4(A-B) = ?— ^cot-|C. 
^^ ^ a + b ^ 

This formula will be used in a later chapter. 



104 THE KIGHT-ANGLED TRIANGLE 

Examples V. 

In the following examples : 

A, B, C are the angles of a triangle ABC. 

a, b, c are the sides, s = half the sum of the sides ; R is the 
radius of the circumcircle. 

r is the radius of the inscribed circle. 

f\ is the radius of the escribed circle touching the side BC. 

A is the area of the triangle. 

D, E, F are the middle points of the sides BC, CA, AB, 
respectively. 

X, Y, Zare the feet of the perpendiculars let fall from A, B, C 
respectively on the opposite sides. 

is the centre of the circumcircle. 

/ is the centre of the inscribed circle. 

K is the orthocentre. 

1. Express in terms of the sides and angles the lengths of AX, 
BX, CX, AK, BK, CK. 

2. Express the length of ^i) in terms of (i) a,b,C, (ii) a,b,B, 
(iii) a, b, c. 

3. Show that a/sin A — b/sinB = c/sin C = 2R. Deduce that 
R = abc/iA. 

4. Prove that r (cot IB + cot^C) = a. Write down the two 
similar formulae. 

5. Prove that r = A/s. (No trigonometiy required.) 
Deduce that tan|^ = A-r {s(s-f/)}. 

6. Show that BX = a — ?>cos c ; hence prove that 

c^ = a"^ + b^ — 2 ab cos c. 

7. Prove that (i) A = ^ab sin C, (ii) A = rs, (iii) A = abc-i-iR, 
(iv) A = -v/s(s-«)(s-&)(s-c). 

8. Prove that 



(i) sin^A= ^/{s~b) (s-c) -^bc, (ii) cos^^= \/s{s-a} -r-bc, 
(iii) tan ^A = y/{s -b){s — c)-TS{s — a). 



THE RIGHT-ANGLED TRIANGLE 105 

9. Show that the triangles ABC and A ZZare equiangular ; hence 
prove that YZ = a cos A. 

10. Two tangents are drawn from a point P to a circle of 
radius 10 cm. ; the tangents contain an angle of 43°. Find the 
lengths of the tangents and the distance of P from the centre. 

11. A sheet of iron is shaped so that it can be rolled up to form 
a conical funnel 6 feet high with open circular ends 2 feet and 
6 feet diameter respectively. Draw a plan of the sheet before 
rolling. What is the inclination of the edge of the funnel to the 
line joining the centres of the ends ? 

12. A circle rolls without slipping along a straight line : prove 
that the co-ordinates of a point fixed to the circumference are such 
that x = a (^ — sin 6), i/ = a (1 -cos 6) ; the origin being taken at 
the point where the fixed point meets the straight line, and 6 being 
the angle turned through by the circle. 

13. One of the angles of a right-angled triangle is the acute 
angle whose sine is §, and the length of the shortest side of the 
triangle is 10 feet. Find the lengths of the other two sides. 

14. ^ is the highest point of a sphere with centre ; a particle 
slides from a position P, where the angle AOP = 6, to the position 
Q where the angle AOQ is 0. How much lower is Q than P and 
how much further from OA ? 

15. The time t of sliding from rest down a length s inclined at 
6 to the horizon is given by s = ^^^^sin^ where g is a constant. 
A circle is held with a diameter AB vertical ; prove that the time 
of sliding along a chord from the highest point A to the circum- 
ference is the same whatever be the inclination of the chord, and 
that the time of sliding from the circumference along a chord to 
B is also independent of the inclination of the path. 

16. A plane, inclined at 20" to the horizon, is placed with the line 
of greatest slope pointing north. A line is drawn on the plane, 
pointing NNE. ; find the inclination of this line to the horizontal. 

17. A man 6 feet high walks along a straight line which passes 
3 feet from a lamp-post. If the light is 9 feet from the ground, 
find the length of the man's shadow when his distance from the 
point on his path nearest to the lamp is 10 feet. What is the locus 
traced out by the extremity of his shadow as he walks along the 
line ? 



106 THE RIGHT-ANGLED TRIANGLE 

18. If, in the previous question, there is a vertical wall parallel 
to the man's path and distant 2 feet from it on the side remote 
from the lamp, what is then the length of the shadow and the 
locus traced by its extremity ? 

19. Draw the graph of 6 /sin 6 from ^ = to ^ = i tt. 
Use the graph to solve the following problem. 

A string 30 inches long is tied to the ends of a cane 35 inches 
long, thus forcing the cane into a circular arc. Find the radius of 
the arc correct to the nearest inch. 

20. Find the length of a strap which passes tightly round two 
pulleys of radii 2 feet and 3 feet, their centres being 6 feet apart. 



CHAPTER VI 



THE TKIANGLE 

Several formulae connecting the sides and angles of a triangle 
have been proved in the examples of the preceding chapters. 
They are here gathered together for reference and proofs are 
given. Care should be taken that the proof applies when 
the triangle is obtuse-angled ; if it does not, a separate proof 
must be given. 

Relations between the sides and angles. 
54. The angle formula. A + B + C = 2 right angles. 

a ^ ^ ^ 

sin A ~" sin B "~ sin C 

A 



The sine formula. 



(=211). 





Fig. XXXIV. 

Let be the centre of the circumcircle, and D the middle point 
of^a 

Join OB, OC, OD. 

Then, in the left-hand circle of Fig. XXXIV, 
angle 300 = 2'' angle BAC 
= 2A. 
Triangles BOD and COD are congruent ; (Prop. 8 a.) 

.'. BOD = COD = A. 
Also BD==^BC = ^ a. 



108 



THE TRIANGLE 



In the right-angled triangle BOD, 

BD = OB sin BOD, 
i.e. \a = R^mA\ 

Sin A 
In a similar way it may be proved that 

-X^ = 2E and ^ = 2R. 
sin B sm C 



sin J. sin^ sm C 



Hence 

Exercise. Supply the proof when the angle A is obtuse. 



Note. In using this formula the following algebraic result is often 
useful : 

If - =z - ^ - then each fi*action equals ; :.. 

55. The cosine formula cos A = — ^r^. , and its 

2 be 

equivalent a^ = b^ + c^— 2 be cos A. 

This can be proved very shortly by assuming Euclid II, 13 and 14 ; 
but it is better to base the proof on the theorem of Pythagoras. 





Let CZ be the perpendicular from C on AB, Fig. XXXV. 
ThenZC= &sin^, AZ=b cos A, and BZ=c-b cos A. 
BC^ = BZ^ + ZC\ 
a' = {c-h cos Af + (b sin Af 

= c'^-2bccosA + b^cos^A + b'^sm'^A, 
i.e. a'' = bUc'' -2 be cos A, 

h-i + c^-a' 



or 



cos^ = 



2 he 



THE TKIANGLE 109 

If ^ is obtuse, then in Fig. XXXV a, 

ZC = & sin (180-^) = &sin^, 

^Z=Z>cos(180-^) = -&cos^, 

BZ= BA + AZ= C + {-bcosA} = c-hcosA. 

The proof is now the same as before. 

Exercise. Write down the corresponding formulae for cos B 
and cos C. 

56. The Projection formulae 

c = b cos A + a cos B. 

In Fig. XXXV, BZ is the projection of BC on BA ; and AZ is 
the projection of ^C. 

AB = AZ+BZ, 
i.e. c = hcosA + a cosB. 

Exercises. Supply the proof when A is obtuse. 
Write down the other two corresponding formulae. 

57. Area formulae 

The symbol A is used to denote area of triangle. 

(i) A = ^ any side x perpendicular from opposite angle. 

(Prop. 16.) 
(ii) A = ^ AB X ZC = ^ c . b sin A = ^ be sin A. 

(iii) A = ^/3 (s - a) (s - b ) (s - c). 

In Fis. XXXV. BZ ^ acosB =a X '-±!! "- . 

2ca 

.'. (2c.Z(7)2 



= {a-\-h + c){a — b + c){a + h — c) -h + c — o). 

Let 2s= a + b + c, then h-^c — a = 2(s — a) &c. ; so that 
2c . ZC= ^/2s.2(s-a). 2 (s - &)T2 (s-c) ; 
A =iAB.ZC 



= >/s(s-a)(s-b)(s-c). 
Exercise. Show that 

16 A2 = 2 (&2c2 + c-a^ + a' h") - (a* + h' + c'). 



no 



THE TRIANGLE 



58. From these formulae others may be deduced. 

Example i. To show that in any triangle 

cos (A + B) = cos AcosB — sin A sin B. 
From sine formula «sin ^ — &siii A = 0. (i) 

From projection formula a cos B + h cos A = c. (ii) 

Square and add, a"^ + &" + 2 cch (cos A cos -B — sin A sin B) = c^ 
From cosine formula a'^ + h'^ — 2 ah cos C = c^ 

It follows that 

cos C = — (cos A cos ^ — sin ^ sin B). 
From the angle formula C = 180 -(^ + -5), 

i.e. cos C= — cos (^ + jB). 

Hence cos (A + B) = cos AcosB-~ sin A sin B. 

Example ii. In any triangle 

sin {A — B) = silt A cos B — cos A sin B. 
Multiply together equations (i) and (ii) above. 
a^ sin ^ cos 5 - h"^ sin ^ cos yl - ah (sin ^ cos 5 - cos A sin B) = 0. 
From Fig. XXXVI it is seen that 
«2 sin B cos B = BZ.ZC= 2'' triangle BZC, 
and h"^ sin ^ cos ^ = 2*^^ triangle AZC 

= 2'Uriangle .4'ZC, 
{ZA'= ZA, so that triangles CZA, CZA' are congruent). 
.-. ci^ sm B cos B -W sm A cos A = 2 ''triangle 5C/1' 

^BC.CA' sin BCA' 
= ahsui{A — B). 




Comparing this with the result above, we see that 
sin {A-B) = sin AcosB- cos A sin B. 
This result can, however, be obtained more quickly. 



THE TRIANGLE 



111 



For 



I.e. 



sin BC A' 

~ZrB~ 

sinjA-B) 
a cos B — b cos ^1 



sin CBA' 
sin^ 



„ • / < T,N « sin 5 cos 5 < • D 

Hence sin (^ — 5) = cos^smJ? 

= sin A cos B — cos A sin B 
since a sin B = h sin A 

Example iii. To show that the area of a quadrilateial inscribed 
in a circle is \/{s — a) {s -b){s — c){s — d) where s — lUi + b + c + d). 
In Fig. XXXVII 

Area of ABCD = sum of triangles ABD and BCD 
= I ad sin A + ^lc sin {180 -A) 
= I (ad + bc) sin A. 
From triangle ABD, 

BD"' = a"^ -\- d"^ -2 ad cos A. 




Fig. XXXVII. 
From triangle BCD, 

BD'' = b'' + c'' -2 be cos {180 -A). 

Hence a^ + d'^-2 ad cos A = h"^ + c"- + 2 be cos A, 

i.e. 2{ad + bc)cosA = a'' + (P-{b''-hc''); 

2{ad + bc){l + cosA) = {a + dy-{b-c)'', 

and 2(((d + bc){l-cosA) = {b + c)''-{a-d)\ 



112 THE TRIANGLE 

Hence i {ad + bc)^ {1 - cos"^ A) 

= {-a + b + c + d){a-h + c + cl)(a + h-c + d){a + b + c-d), 
i.e. {i(af? + &c)siii^}2 

= I {- a + b + c + d) I {a-b + c + d) l{a + b - c + d) ^{a + b + c- d) 
.'. Area of ABCD = ^/{s-a){s-b){s-c){s-d}. 

Examples Via. 

1. From the three projection formulae deduce the three cosine 
formulae. 

2. Prove that sin^ = sinPcos C+ cos^sin 0; and deduce that 
sin (B + C) = sin BcosC + cos B sin C. 

3. Prove that cos(J.-5) = cosvl cos5 + sin^sin5. 

4. Show that A = |- (&^ sin CcosC + c^ sin B cos B). 

5. Show that A = i c^ {sin AsinB-^ sin (^ + ^)} . 

6. Prove that sin A + sinB> sin C. 

7. Prove that cot yl + cot5 = ccosec-B-^a. 
What third expression are these equal to ? 

8. Show that 

R (i.e. the radius of the circumcircle) = s -^ (sin ^ + sin ^ + sin C). 

9. Use the formula cos^ = l-2sin'^|^ to prove that 

sin 1^ = \/(s -b){s-c)-r- be. 
Write down the similar formulae for sin J 5 and sin | C. 

10. In a similar way to that suggested in the previous example, 
prove that cos |^ = ^/s {s-a) -f be. Write down the formulae for 
cos 1 5 and cos I C. What is the formula for tan^yl ? 

11. Given cf = 17, 5 = 12, 5^= 37° 15', find A. 

12. Given a = 14, b = 13, c = 12, find the greatest angle. 

13. Given a = 45, A = 45°, B = 60°, find b. 

14. Given b=ll, c = 42, A = 72°, find a. 

15. Given a = 176, b = 291, c = 352, find all the angles. 
(Choose a formula adapted for logarithms.) 

16. Given « = 7, fc = 5, C = 49°, find e. 

17. Given b = 9,c=10,C= 57°, find a. 

18. By considering two forms for the area of an isosceles triangle, 
prove that sin ^ = 2 sin J A cos | A. 



t 



THE TRIANGLE 113 

19. Two sides of a triangle are 3 and 12 and the contained angle 
is 30^ ; find the hypotenuse of an isosceles right-angled triangle of 
equal area. 

20. Two adjacent sides of a parallelogram, 5 inches and 8 inches 
long respectively, include an angle of 60°. Find the length of the 
two diagonals and the area of the figure. 

21. If in a triangle C = 60^, prove that 

l/(a + c) + l/{b + c) = 3/(rt + b + c). 

22. On a straight line AB, 4 inches long, describe a semicircle, 
and on the arc of the semicircle find points P, Q, i?, S such that 
the areas of the triangles APB, AQB, ARB, ASB are 1 square inch, 
2 square inches, 3 square inches, and 4 square inches respectively. 
If C is the centre of the circle, determine the sines of the angles 
ACP, ACQ, ACR, and ACS, and hence find, from the tables, the 
values of these angles. 

23. If a quadi-ilateral can be inscribed in one circle and circum- 
scribed about another, show that its area is ^/ctbcd, where a, h, c, d 
are the lengths of the sides. 

The circles of the triangle. 

59. It is shown in any Geometry textbook that 
(i) the centre of the circumcircle is the point of concurrence of 
the perpendicular drawn at the middle points of the sides ; 

(ii) the centre of the inscribed circle is the point of concurrence 
of the three lines bisecting the three angles ; 

(iii) the centre of an escribed circle is the point of concurrence 
of the bisector of the opposite interior angle with the bisectors of 
the two adjacent exterior angles. 
In Fig. XXXVIII, we have 

AQ = AR, (Prop. 24.) 

BP = BR, 
CP=CQ; 
.'. AQ-\-BP+CP=^. sum of sides = s. 
Hence AQ = s-a. 

Exercise. In a similar way, prove that 
BP = CF = 

CQ = QQ' = 

AQ' = PF = 

1216 H 



114 



THE TRIANGLE 




Fig. XXXVIII. 



Examples VI b. 

Prove the following formulae : 

\. K = a^2&mA. 2. 7? = rt/;c-r4 A. 

3. r = A/s. (Considerthesum of the triangles P/C, (7/.4,yi/B.) 

4. r = rt^(cot^i? + cot|C). 5. ;-j = A/(.s-cf). 
6. ri = rt^(tanJi?-t tan ^ C). 



THE TEIANGLE 115 

Using the above formulae, prove the following relations ; 

7. In a right-angled triangle R + r = ^{a + h). 

8. l//'j + 1/^2 +1/^3 = 1/r. 9. l/r^+l/r^ = 2 -=-6 sine. 
10. r7\ror2,= ^". 11- r r-^ = {s-h){s-c). 

12. (aZ^c-fsin J.sin^sinC)^. 13. 27?/- = «&c-f(a + & + c). 

14. 4i? sin yl sin B sin C = a cos ^ + 6 cos B + c cos C 



15. tanf^ = V'(s-6j(s- c)-rs(s- a). 

16. s2= AcotMcotii^cotiC. 

17. If ABC is a triangle such that 2h = a-\-c, and ^ is the 
length of the perpendicular from B upon AC, show that tan | J. 
and tan \ C are equal to the roots of the equation 

18. Show that the sum of the radii of the escribed circles of 
a triangle is equal to the radius of the inscribed circle together 
with four times the radius of the circumscribing circle. 

19. Show that the area of the triangle formed by joining the 
centres of the escribed circles is 

Si^^cos-l^cosi^cos^O. 

20. The sides of a triangle are 3, 5, 6 ; find the radii of the inscribed 
and circumscribed circles. 

21. In an isosceles triangle the base is 100 cm. and the perpen- 
dicular from the vertex is 70 cm. ; find the radii of the inscribed 
and circumscribed circles. 

22. A triangle is described with base BC = 5 inches and angle 
A = 70°. What is the radius of the circumcircle ? Find the dis- 
tance of the centre of the circumcircle from BC. 

23. Find the radius of the circumcircle of the triangle ABC being 
given that BC = 7, CA = 6, and C = 60°. 

24. If a = 32, & = 16, C = 42°, find R and r. 

25. The area of a parallelogram having base 5*8 cm. and angle 
123° is 37*7 sq. cm. Find the other sides and angles. Find the radii 
of the circles which pass through three of the corners of this 
parallelogram. 

26. Two of the sides of a triangle are 7'5 cm. and 9'3 cm., the 
included angle is 37'. Find the radius of the circle which touches 
these sides produced and the third side. 

h2 



116 THE TRIANGLE 



Oral Revision Examples. 

Complete the following identities and equations : 

1. sin (270-^)- 2. cos2(9 = 

3. 2 tan ^ cot ^ = 4. If sin 6 = 1,0 = 

5. In any triangle &^ = 6. In any triangle R = 

1. sm2A= 8. tan 225° = 

9. If cos ^ = #, tan 6 = 10. A in terms of the sides = 



11. tan-il = 12. secM-1 = 

13. length of arc = radius x 14. sin'^5 + sin^(90 — J5) = 

15. Definition of tangent. 16. In any triangle cosC' 

17. In any triangle h cos C+c cos B = 

18. In any triangle &csin^ = 19. In any triangle r = 

20. tan I TT = 

21. Definition of sine. 22. cos (360° -jB) = 

23. In any triangle r^ = 24. In any triangle cos^ 

25. What formula connects a, b, and B? 

26. tan-'(--s/3) = 27. If cos/?= -^6 = 
28. tan2 73l° + l= 29. abc = 

30. 37° = ? radians. 



31. cos2(^-45°) + sin2(^-45°) = 

32. &sinC = 

33. Express R in terms of the sides. 

34. If sin (9 = sin 0^, then 6= 35. cos^= (in terms of sin |^), 
36. Area of triangle = 37. aH c^ - 2 «c cos B = 

38. cos 1200° = 

39. Maximum value of 2 sin ot cos a = 

40. cos^^-sin*^ = (in its simplest form). 



41, A -r (.<;-«)= 42. tan (180-5) = 

43. acos C + CC0S.4 = 4t4:. he sin A = 

45. sui^ {A + B) + cos"^ {A + B) = 46. If cos^ = cosyl, then x = 

47. In any triangle cos A = 48. tan 60° = 

49. cos 2 ^ = 50. How many radians = >4° ? 



THE TKI ANGLE 117 

Examples VI. 

1. Prove that (a cos A - h cos B) -r (a^ - h') + cos C/c — 0. 

2. Prove that c- = {a + hf sin- 1 C + {a- h)- cos- }, C. 

3. In a triangle ^i?Cthe lines drawn from A and C, perpen- 
dicular to the opposite sides, intersect in 0. If the angle A is 
acute, show that OA == b cos A/sin B. 

Also draw a diagram in which A is an obtuse angle, and establish 
the corresponding expression for OA in that case. 

4. Show that in any triangle the product of a side and the sines 
of the two adjacent angles is the same, whichever side be taken. 

5. Find the area of a regular polygon of n sides circumscribed 
about a circle of radius r. 

6. Regular polygons of 1 5 sides are inscribed in and circumscribed 
about a circle whose radius is one foot ; show that the difference 
of their areas is nearly 20 square inches* 

7. ABCD are four points on a circle such that the angles BAC 
and BCA each equal 6. Show that AD +CD = 2BD cos 6. 

8. If 2 cos B = sin A/sin C, prove that the triangle is isosceles. 

9. If tan.-l/tan^ = sin-.-1/sin-i?, show that the triangle is 
isosceles or right-angled. 

10. Express the sides of a triangle in terms of the angles and the 
semi-perimeter. 

11. In a triangle ABC perpendiculars AD and BE are let fall on 
the opposite sides ; prove that the radius of the circle circum- 
scribing the triangle CDE equals R cos C. 

12. If in a triangle the median bisecting the base AB is perpen- 
dicular to the side AC, prove that 2 tan A -f tan C = 0. 

13. If ^; and q are the lengths of the perpendiculars from A, B on 
any arbitrary line drawn through the vertex of a triangle, prove 
that a'p"" -f &' 5' - 2«Z> pq cos C = w^ ^2 ^^^2 q 

14. An isosceles triangle, vertical angle 35°, is inscribed in a circle 
whose radius is r65 inches. Find the lengths of the sides. 

15. Show that in any triangle 

cos A cos B cos C ^ a^ ■\-}r A- (? 
a b c 2abc 



118 THE TRIANGLE 

16. If R is the radius of the circumcircle of any triangle and 
X, y, z are the lengths of the perpendiculars let fall from its centre 
on the sides, prove that 

R^-{x^ + y'^ + z')R-2xyz = 0. 

17. The rectangular co-ordinates of the angular points of a triangle 
are (4, 5), (6, 7), (8, 6) ; determine the sum of the two smaller 
angles. 

18. A rod AB, length 2 a, can turn about a hinge fixed to the 
wall at ^ ; it is supported by a string BC, length /, fastened to 
a point C on the wall at a height h above A. 

(i) If BC is horizontal, what is the inclination of the rod to 
the vertical ? 

(ii) If BC is horizontal, what is the inclination to the vertical 
of the line joining the hinge to the middle point of the string ? 

(iii) If the string and rod are inclined at 6 and to the vertical 
respectively, prove that (i) 2 a sin = Z sin 6, (ii) J cos 6 — 2 a cos (^ = /i. 

(iv) In the general case, what is the angle between the string 
and the rod ? Give the answer in terms of li, a, 6 or li, I, 0. 

(v) In the general case, what is the inclination to the vertical 
of the line joining the hinge to the middle point of the string? 
Give the answer in terms of h, a, (p. 

19. Three equal spheres of radius 7 centimetres are fixed in 
a horizontal plane so as to touch each other ; a sphere of radius 
6 cm. rests upon these three. Find the height of the centre of the 
fourth sphere above the horizontal plane, and the inclination to 
the vertical of the line joining the fourth centre to one of the 
lower centres. 

20. Three equal rods of length 54 inches are fixed so as to fonii 
a tripod. If their feet are at the corners of an equilateral triangle, 
side 18 inches, find the inclination of each rod to the vertical. 

21. In any triangle prove that the centroid trisects the line 
joining the circumcentre to the orthocentre. 

22. Find the lengths of the sides of the pedal triangle of the 
triangle ABC. Find also the radii of the inscribed and circum- 
scribed circles of that triangle. 

(The pedal triangle is formed by joining the feet of the perpen- 
diculars let fall from the vertices on the opposite sides.) 

23. If rt = 5 and ft = 4, draw a graph to show the value of c as 
C varies from 0° to 180°. Hence find the value of c when C = 40^ 



CHAPTER VII • 

SOLUTION OF TRIANGLES 

60. It is known from Geometry that, if three parts of a 
triangle are given, the remaining parts can in some cases be 
found; and that, in other cases, relations between the missing 
parts may be found even though their exact values cannot be 
determined. When actual numbers are given, results can be 
obtained to a greater degree of accuracy by Trigonometrical 
methods than by drawing to scale. In all cases a formula is 
sought which shall contain the three given letters and one 
unknown letter. 

61. Case I. Three angles given. 

The angle formula shows that A + B+C must be 180°. No 
formula contains the three angles and one side only ; but from 

the sine formula, viz. -. — - = -. — ., = -. — -, we can find the ratios 
sm ^-1 sin B sm C 

of the sides. 

62. Case II. Two angles and one side given. 

The third angle can be found immediately since 
A + B+C= 180". 
Suppose a is the given side ; and it is required to find b. The 
formula must contain a, h, and two of the angles ; hence we use 
b _ a 
sin B sin A 
This is adapted for the use of logarithms as it involves no 
addition or subtraction. If the tables in use give the logarithms of 
the cosecant, it may be advisable to use the following logarithmic 
form log b = log a + log sin B + log cosec A. 

63. Case III. One angle and the two sides containing 
the angle are given. 

Suppose a, b, C are the given parts. Then the cosine formula 
c^ = ci^ + b'^ — 2 ab cos C enables us to determine c. When c is 



120 SOLUTION OF TRIANGLES 

determined, the remaining angles can be found by the sine 
formula. 

This method is of practical use only when the numbers involved 
are small ; the cosine formula is not adapted for the use of 
logarithms. It is usual, therefore, to use the formula proved in 

§ 53,* viz. tan 1{A-B) = ^^ cot I C. 

This determines h{A—B)', also ^{A + B) equals the complement 
of I C; hence A and B are found by adding and subtracting. 
The value of c is then calculated by the sine formula. 

64. Case IV. One angle and the two sides not 
containing the angle are given. 

Suppose a, h, A are given. Then we can determine c from the 
formula a^ = b'^ + c'^ — 2bc cos A. 

This is a quadratic equation to determine c, and it is seen that 
there is the possibility of two distinct values for c. This is also 
seen from the geometrical construction. On this account this 
case is usually known as the Ambiguous Case. 

If there are two values of c, there will be two values for B and 
for C. This is seen independently if the sine formula is used 
(as it usually is, on account of its adaptability for logarithms) : 
sin B _ sin A 
b a 

Suppose that this leads to the result 
sin B = sin x. 

Then B = x or 180 -ic. 

This shows that, if there are two solutions, those two solutions 
are supplementary. Hence one of the solutions will be obtuse. 
Preliminary geometrical considerations often show that there can 
be only one solution. 

(i) If the given angle A is not acute, then B must be acute and 
the obtuse-angled solution must be rejected. 

(ii) If a>b or = h, then A> B or = B; consequently B 
cannot be obtuse. 

Exercises. When a, b, A are given, show (i) from the geo- 
* Another proof is given on p. 103. 



SOLUTION OF TKIANGLES 121 

metrical solution, (ii) from the cosine formula, (iii) from the 
sine formula, that 

(a) there is no solution, if a<h sin A ; 

(6) there is one solution only, if a = & sin ^ ; 

(c) there are two solutions, if a>68in^ but < &; 

[d) there is one solution only, if « > &. 

Point out the difference in nature of the one solution in {h) 
and [d). 

65. Case V. Three sides given. 

Here again the cosine formula may be used, if the numbers 
involved are not inconveniently large. For logarithmic calcula- 
tion the formula for sin^^, cos|^, or tan|^l is used. These 
half-angle formulae are derived from the cosine formula. 
2bin2M = l-co8^1* (§51) 

~~ 2bc 



2 he 
{a — b + c){a + b-c) 



ainiA- j (s-b)(B-c) 

srn.A-^ 



2 



Similarly cos ^ A = \ ~^ — ' 



Divide tan^A=J^^:^l^ 

^ \ s(s-a) 



Of these three formulae it is best to use the tangent formula ; for 
the logarithms used in finding tan 4 A are the same as those required 
for finding tan i B or tan I C. If only one angle lias to be found, it is 
indifferent which formula is used. 

There is a simple geometrical proof for tan \ A. 

* In old books on Trigonometry the ' haversinc ' was used for 
solving triangles, and the values of log haversinc were tabulated in 
mathematical tables. The haversinc equals half the versed sine ; 
hence haversin -4 = J versin A = (1 — cos^)-^2 = sin^ \ A. The for- 
mula for solution of the triangle then becomes 

havei'sin A = (s—b) (s — c)-T-bc. 



122 SOLUTION OF TRIANGLES 

In Fig. XXXVIII / is the centre of the inscribed circle, £" is the 
point of contact of the circle with AC. 

Then tan ^ = 42 



2 AQ 
r 
s~a 
A 



(§59) 



(s-rt) 



ls -h){s-c) 
\ s{s — a) 

Examples Vila. (See p. 81 for arrangement of work.) 
In the following triangles when 
Case I. 

1. A = 79° 20', B = 64" 10', find the ratios of the sides. 
Case II. 

2. Ar= 58° 12', B = 64° 33', a = 385, find b. 

3. ^ = 38° 24', C = 95° 5', c = 7-832, find a and b. 

4. 5 = 63° 55', C = 48°27', c = b'16, find 6. 

Case III. 

5. a = 409, b = 381, C = 58° 12', find A and B. 

6. B = 23° 46', c = 9-72, a = 8*88, find A and 0. 

7. a = -532, c = '259, B = 39° 33', find A and C. 

8. ^ = 73° 15', b = 7315, c = 8013, find B and 0. 

Case IV. 

9. A = 38° 14', a = '33, 6 = '44, find C. 

10. « = 409, b = 385, A = 64° 32', find B and C. 

11. 6 = 6-901, c = 5-749, 0=48° 27', findi?. 

12. A = 73° 15', a = 7315, c = 8013, find B and C. 

Case V. 

13. a = 17, fe = 13, c = 12, find the least angle. 

14. a = 793, b = 937, c = 379, find all the angles. 

15. s = 1410, a = 1437, b = 811, find all the angles. 

16. s = 1437, a = 1410, b = 811, find all the angles. 



SOLUTION OF TRIANGLES 123 

17. ft = 13, h = l, C= 60^ find A and B. 

18. a = 32, h = 40, c = 66, find C. 

19. a = 250, Z) = 240, A = 72° 4', find B and C. 

20. a = 2 &, O = 120°, find ^, 5 and the ratio of c to a. 

21. rt = 86, & = 63, c = 81, find the smallest angle. 

22. & = 5, c = 3, ^ = 42°, find B and C. 



Oral Examples. 

State the formula to be used in the following cases : 

1. Given a, h, C, find c. 2. Given a, h, C, find A and B. 

3. Given h, c, C, find B. 4. Given b, c, C, find «. 

5. Given c, a, C, find A. 6. Given c, a, C, find 5. 

7. Given c, A, B, find C. 8. Given c. «, A, find &. 

9. Given a, h, B, find C. 10. Given a, 5, ^, find c. 

11. Given «, b, c, find C. 12. Given A, B, C, find a. 

13. Given ^, C, b, find a. 14. Given a, b, B, find ^. 

15. Given a, c, B, find C. 16. Given c. A, B, find &. 

17. Given «, fe, r, find B. 18. Given &, c, A, find £. 

19. What is the ambiguous case? 

20. When a, c, A are given, what are the conditions that there 
should be no ambiguity? 

Examples VII b. 

Solve the following triangles : 

1. a = h,b = l, C=30°. 

3. a = 65, & = 68, c = 16. 

5. a = 7, ^ = 120°, A = 45°. 

7. 6 = 926*7, ^ = 48° 24', B 

S. a = 407-4, c = 115'9, A = 127° 45' 

9. rt = 1263, b = 1359, c = 1468. 

10. a = 53-94, b = 156-5, C = 15° 13'. 

11. b = 457-2, c = 342-6, A = 73° 45'. 

12. rt = 246-7, Z; = 342-5, B = 32° 17'. 

13. c = 79-48, A = 54° 16', B = 85° 6'. 

14. rt = 7-956, b = 10-35, c = 9-412. 

15. b = 9463, c = 7590, C = 43° 47'. 

16. a = 739, c = 937, £ = 146° 12'. 



2. 


b = 


4, 


c = 


= 3, 


C = 


= 60° 


4. 


b = 


8, 


c = 


= 9, 


c = 


:45°, 


6. 


a = 


= 6, 


& = 


= 7, 


c = 


5. 


31' 


= 13' 













124 



SOLUTION OF TKIANGLES 



17. c = 79-5, A = 35° 14', C = 117° 35'. 

18. A = 89°, B = 18° 47', C = 72° 13'. 

19. a = 87-6, b = 57'4, c = 46*8. 20. a = 79, c = 97, A = 2437. 
21. A = 79°, C = 97°, R = 17-2. 22. b = 73-6, R = 57, a = 48*9. 

23. ft2 + &2^34i^ sinC= 1, tanJ5 = fQ. 

24. ^ = 42° 35', a = 83, b = 74. 

25. a = 2-740, b = '7401, C = 59° 27'. 



Heights and Distances. 

66. First a figure must be drawn, not necessarily to 
scale ; the known lengths and angles should be indicated 
in the figure. It may be necessary to solve, or partly solve, 
more than one triangle before the required measurement is 
found. The scheme for working should be carefully thought 
out before the work is actually begun. 

Example i. Wishing to find the height of a house standing 
on the summit of a hill of uniform slope, I descended the hill 
for 40 feet, and then found the height subtended an angle of 
34° 18'. On descending a further distance of 60 feet, I found 
the subtended angle to be 19° 15'. Find the height of the 
house. 




Fig. XXXIX. 

Scheme.— In triangle ADC we know one side CD and all the angles ; 
so AC can bo found. Then in the triangle A CB two sides AC, CB aro 
known, and the included angle, hence AB can be found. 



HEIGHTS AND DISTANCES 



125 



From triangle ACD, 
AC 



CD 



sin ADC sin DAC 
i.e. ylC = 60sin 19^ 15' cosec IS*" 

= 76-182. 
From triangle ABC, 

h — a 



3' 



tan ^(B-A) = , cot i C 

36-182 



Again, 



116-182 
^{B-A) =45° 16', 
i{B+A) = 12°bV- 
A = 27° 35'. 

AB _ ^inACB 
'CB ~ sin cab' 
40 sin 34= 



cot 17^ 9' 



Logarithms. 
1-77815 
1-51811 
-58559 

1-88185 



1-55849 
+ -51061 



AB = 



18' 



2-06910 

2-06514 

-00396 



1-60206 
+ 1-75091 



35' 



1-35397 
1-66562 



1-68835 



sin 27' 

= 48-792. 

Height of house = 48-8 feet. 

Example ii. Wanting to Icnoiv the height of a castle on 

a rock, I measured a base line of 100 yards, and at one 

extremity found the angle of elevation of the castle's top to he 

45° 15', and the angle subtended hy the castle's height to he 

34° 30'; also the angle subtended by the top of the castle 

and the oilier extremity of the base line was 73° 14'. At 

the other extremity the angle between the first extremity and 

tJie top of the castle teas 73° 18'. Find the height of the castle. 

This requires a rough perspective figure of the whole, and sub- 
sidiary plane figures. 

A 




Fig. XL. 



126 



HEIGHTS AND DISTANCES 



AB represents the castle. 

C is the point in the same vertical as AB, and in the same 
horizontal plane as DE, the base line. 

The following magnitudes are known : 

DE = 100 yards. 

ACD and ACE are each right angles. 

ABC, ABB are known, therefore BBC is known. 

ABE, AEB are known. 

Scheme. In triangle ADE, DE and the adjacent angles are known ; 
hence AD can be found. AB can now be found from triangle ABD. 

A 



From triangle ABE, 

AB BE 



I.e. 



sin 73° 18' 
AB 




2- 

+ 1-9813 
- 17415 



log^Z) = 2-2398 



Fig. XLII. 



From triangle ABB, 



AB 



.ID sin 34° 30' 



sin 79° 15' 
Height of castle = 100 yards. 



2-2398 
+ 1-7531 

1-9929 
-1-9923 



2-0006 



HEIGHTS AND DISTANCES 



127 



Example iii. From the top of the Feak of Teneriffe the 
dip of the horizon is found to he 1° 58'. If the radius of 
the earth be 4000 miles, ivhat is the height of the mountain? 

In Fig. XLIII C is the centre of the earth, AB is Teneriffe ; 
BH is the tangent drawn from B to the earth's surface, so that H 
is the farthest point seen from B ; in other words, H is on the 
horizon. The angle between BH and BD (the perpendicular to 
the vertical) is called the dip of the horizon. 




Fi;?. XLIII, 



From triangle BCH, 



BC^ 
CH 



sec BCH, 



angle BCH = complement of CBH = HBD ; 
i?C = 4000 sec 1°58' 
= 4000x1-00059 
= 4002*36 miles. 
Height of mountain is 2*36 miles. 



Note. Fig. XLIII is drawn much out of scale ; for small heights BH 
and BD are practically identical. Even for mountains the dip is very 
small, as in this example ; in fact, so small that we may use the 
approximation sine of dip = tan of dip = circular measure of dip. 



128 HEIGHTS AND DISTANCES 

If E be the other extremity of the diameter through B, we have, 
from § 52, 

BA. BE = Bir\ 

i.e. h{2r + h)=- ct", 

whore r is radius of earth, h is height of place of observation, d is 
the distance of the horizon. 

Hence d = \/2 rh-\-h'^', 

= -s/2 rh ( \ + — — — ^... ) by the Binomial Theorem. 
\ 4 r 32 r' / 

So far the work is accurate ; usually h/r is so small that it may be 
neglected. Hence for ordinary heights 

Distance of horizon = v 2 rh. 

Exercise, (i) In the formula just obtained r, /;, and the distance 
are all expressed in the same units. By taking r = 3960 miles, prove 
that 
Distance of horizon in miles 

= •v/f X height of place of observation in feet. 
(ii) Show also that 

Dip in minutes = '9784 V height in feet. 



Examples VII. 

1. Standing at a horizontal distance 100 yards from the foot of 
a monument, a man observes the elevation of its top to be 25° 35'. 
Assuming the man's eye to be 5 feet from the ground, find the 
elevation of the top when the man stands 50 yards from the foot. 

2. OA and OB are two straight roads intersecting at and 
making with each other an angle of 85° 12'. vl is a house 
1572 yards from 0, and 5 is a house 1129 yards from 0. Find the 
direct distance between A and B. 

3. A man observes the angles subtended by the base of a round 
tower at three points A, B, and C, in the same horizontal straight 
line with the centre of the circular base, to be 2 a, 2 /3, 2 y respec- 
tively. Find the ratio of AB to BC. and find the diameter of the 
tower in terms of ^IC. 



SOLUTION OF TRIANGLES 129 

4. A man observes that the elevation of the top of a tower is 
37^ 40', and that the elevation of the top of a flagstaff on the tower 
is 43= 59'; show that the height of the flagstaff is one-fourth of the 
height of the tower very nearly. 

5. Having given that the least side of a triangle is 17'3 inches, 
and that two of the angles are 63=20' and 72M0', find the 
greatest side. 

6. If two sides of a triangle are 7235 feet and 4635 feet respec- 
tively, and if the included angle is 78=26', find the remaining 
angles of the triangle. 

7. The base of a triangle being 7 feet, and the base angles 
129=23' and 38=36', find the length of the shortest side. 

8. Explain the ambiguous case of the solution of triangles. 
When a, h, A are given and the question is asked whether, from 
these data, two triangles, one triangle, or no triangle can be 
constructed, show that the question can be answered from a 
consideration of the roots of the equation 

x'^ — 2hx cos A-Irl? = c<}. 

9. From each of two ships, a mile apart, the angle is observed 
which is subtended by the other ship and a beacon on shore ; 
these angles are found to be 52° 25' and 75° 10' respectively. 
Find the distances of the beacon from each of the ships. 

10. A ship sailing due north observes two lighthouses bearing 
respectively NE. and NNE. After the ship has sailed 20 miles 
the lighthouses are seen to be in a line due east. Find the 
distance in miles between the lighthouses. 

11. The angles A, J5, C of a triangle ABC are 40°, 60=, and 80° 
respectively, and CD is drawn from C to the base bisecting the 
angle ACB', if AB equals 100 inches, find the length of CD. 

12. A man standing at a certain station on a straight sea-wall 
observes that the straight lines drawn from that station to two 
boats lying at anchor are each inclined at 45° to the direction 
of the wall, and when he walks 400 yards along the wall to 
another station he finds that the angles of inclination are 15° and 
75° respectively. Find the distance between the boats and the 
perpendicular distance of each from the sea-wall. 

13. From a house on one side of a street observations are made 
of the angle subtended by the height of the opposite house, first 

121C I 



130 SOLUTION OF TKIANGLES 

from the level of the street, in which case the angle is tan~^ (3), 
and afterwards from two windows, one above the other, from each 
of which the angle is found to be tan-^( — 3). The height of the 
opposite house being 60 feet, find the height of each of the two 
windows above the street. 

14. A segment of a circle stands on a chord AB 10 cm. long and 
contains an angle of 40°. A point C travels along the arc ; for what 
value of the angle ABC is the chord CA three times the chord CB ? 
Verify by drawing a graph showing the chord CA as a function of 
the chord CB. 

15. If the sides of a triangle are 1011 and 525 feet, and the 
difference of the angles opposite to them is 24°, find (correct to the 
nearest degree) the smallest angle of the triangle. 

16. A ladder is placed against the wall of a room and is inclined 
at an angle Oc to the floor. If the foot of the ladder slips outwards 
from the wall a distance of a feet, and the inclination of the ladder 
to the floor is then /3, show that the distance which the top of the 
ladder will slide down the wall is a cot|(a + /3). 

17. A man travelling due west along a straight road observes 
that when he is due south of a certain windmill the straight line 
drawn to a distant tower makes an angle of 30° with the direction 
of the road. A mile further on the bearings of the windmill and 
tower are NE. and NW. respectively. Find the distances of the 
tower from the windmill, and from the nearest point of the road. 

18. A statue 10 feet high, standing on a column 100 feet high, 
subtends at the eye of an observer in the horizontal plane from 
which the column springs the same angle as a man 6 feet high 
standing at the foot of the column ; find the distance of the 
observer from the column. 

19. It is found that two points, each 10 feet from the earth's 
surface, cease to be visible from each other over a level plain at 
a distance of 8 miles ; find the earth's diameter. 

20. A plane, inclined at 33° to the horizontal, meets a horizontal 
plane in the line BC. From B a line BD is drawn on the inclined 
plane making an angle 27° with the horizontal plane. If BD is 
18 inches long, find the height of D above the horizontal plane, 
and its distance from BC. Also find the angle BD makes 
with BC. 



SOLUTION OF TRIANGLES 131 

2L A lighthouse was observed from a ship to be N. 23" E. ; 
after the ship had sailed due south for 3 miles, the same light- 
house bore N. 12" E. Find the distance of the lighthouse from 
the latter position of the ship. 

22. Two streets meet at an acute angle; the one lies N. 51° W., 
and the other S. 48" \V. The distance from the corner to a 
chemist's door in the first street is 315 yards; and the distance 
from the corner to a doctor's door in the other street is 406 yards. 
Find the length of a telephone wire going direct from the doctor's 
house to the chemist's. 

23. From a vessel at anchor two rocks are observed to the 
westward, the one (A) bearing WNW., and the other (B) W. by S. 
from the vessel. From the chart it is found that A bears NNE. 
from B and is distant 645 yards from it. What are the distances 
of the rocks from tlie vessel ? 

24. Three objects A, B, and C forming a triangle are visible 
from a station D at which the sides subtend equal angles. Find 
AD, it being known that 

AB = 12 miles, AC = ^ miles, CAB = 46" 34'. 

25. A tower on the bank of a river, whose breadth is 100 feet, 
subtends angles 22 1" and 67^" at two points A and B on the 
opposite bank of the river, whose distance apart is 6C0 feet, on 
a level with the base of the tower. Find the height of the tower. 

*26. A, B, C are three given stations, so that the triangle ABC 
is completely known. Show how to determine, by means of 
angles measured at a fourth station P, the distances PA, PB, PC, 
the four stations being all in one plane, the case for considera- 
tion being that in which P is within the angle A, and the points 
P and A on opposite sides of BC. 

If ABC is equilateral, and the angle BPC equals 60', show 
that 2 cos {(^0^ + BAP) + cos (ABP- BPA) =0. 

27. A tower stands on the edge of a circular lake ABCD. The 
foot of the tower is at D, and the angles of elevation of the top of 
the tower from A, B, C, are 0(, ,3, y respectively. If the angles 
BCA, BAC be each equal to 6, show that 

cotan y. + cotan 7 = 2 cotan 3 cos 9. 

* This example is best solved by using the formulae of §§ 83 and 84. 

*i2 



132 SOLUTION OF TRIANGLES 

28. A mountain is observed from a place A to have elevation 
15° 17' and to bear N. 24°29' W. From another place B which is 
2347 yards north of A its bearing is N. 37° 2' W. Deduce the 
elevation from B. 

29. The extremity of the shadow of a flagstaff 6 feet high, 
standing on the top of a regular j^yramid on a square base, just 
reaches a side of the base and is distant 56 feet and 8 feet from the 
extremities of that side. If the height of the pyramid be 34 feet, 
find the sun's altitude. 

30. A man observes that when" he has walked c feet up an 
inclined plane the angular depression of an object in the horizontal 
plane through the foot of the slope is (X ; and that, when he has 
walked a further distance of c feet, the angular depression of the 
object is ^. Show that the inclination of the slope to the horizon 
is cot-^ (2 cot /3 — cot 0() ; and determine the distance of the object 
observed from the foot of the slope. 

31. A straight flagstaff, leaning due east, is found to subtend an 
angle 0( at a point in the plain upon which it stands, a yards west 
of the base. At a point h yards east of the base, the flagstaff sub- 
tends an angle /3. Find at what angle it leans. 

32. Four rods are loosely jointed at their extremities to form 
a parallelogram with sides 4 and 5 inches long. Two of the 
opposite corners are connected by an elastic string of length 
7 inches. Find the angle between the string and the shorter side. 

If the length of the other diagonal be diminished by 1 inch, 
what does the angle become ? 

*33. Three posts on the border of a lake are at known distances 
from each other, namely 63 yards, 44 yards, and 76 yards. At 
a boat on the lake it is found that the two posts, whose distance is 
63 yards, subtend an angle 89° 15', and the two posts, whose 
distance is 76 yards, subtend an angle 130° 45'. Find the distances 
of the boat from the three posts. 

34. A base line AB is drawn 2 chains in length on a plane 
in the same horizontal plane as C the foot of a tree. The angles 
ABC, BAC are found to be 79° 56' and 78° 18' respectively; the 
angle of elevation of the top of the tree is found to be 19° 46' at A. 
Find the height of the tree to the nearest foot. 

* This example is best solved by using the formulae of §§ 83 and 84, 



SOLUTION OF TRIANGLES 133 

35. A base line AB, 2527 links long, is measured on the sea- 
shore along the high water mark. is a point where a distant 
rock meets the sea ; the angles BAC, ABC are found to be 89" 15', 
86" 21' respectively. The angle of elevation of the highest point 
of the rock, which is vertically above C, as observed at A, is 1°48'. 
Neglecting the curvature of the earth, find the height of the rock 
and its distance from A. 

36. A hill slopes upwards towards the North at an inclination 
14" to the horizontal. The sun is 15° W. of S., at an altitude 
of 47° ; find the length of the shadow cast on the hill by a vertical 
post 39 feet high. 

37. If, in the previous question, the post is perpendicular to the 
surface of the hill, what is the lenorth of the shadow ? 



134 REVISION EXAMPLES 

Revision Examples B. 

1. Define the tangent of any angle, and prove from the defini- 
tion that (i) tan(90 + ^j= -cot^; (ii) tan(180-.4)= -tan^. 

Express the other trigonometrical ratios in terms of the tangent. 

2. Show by substitution that 

sin 45" + sin 30° > sin 60°, 
and cos 30° - cos 45° < cos 60°. 

3. Find the value of sin 45° without using tables. 
Solve the equation 4 sin ^ cos ^ -f 1 =2 (sin 6 -h cos S). 
Give the general solutions. 

4. A man walks directly across the deck of a ship, which is 
sailing due North at 4 miles an hour, in 12 seconds, and finds that 
he has moved in a direction 30° East of North. How wide is 
the deck? 

5. Show that in any triangle ABC, 
(i) sin A /a = sin B/b = sin C/c ; 

(ii) sin C {a cos B-b cos ^) = (a + b) (sin A — sin B). 

6. Prove geometrically that 

cos 2^ = l-2sinM. 
Hence find the value of sin 15°. 

7. The angle of elevation of the top of a spire seen from A 
is 30°, and it is found that at a point B, 115| feet nearer the foot 
of the spire, it is 60°. Find the height of the spire to the nearest 
foot. 

8. Plot a curve giving the sum of 4sin^ and 3 sin 2^ from 
^ = 0° to ^ = 180° ; and read off the angles at which the greatest 
and least values respectively of this sum occur. 

Estimate the slope of the curve when 6 = 90° and when ^ = 135°. 



9. Define a radian. Express in degrees and minutes an angle 
of 1'36 radians. 

Find the number of radians in the angle of a regular decagon. 

10. Prove 

(i) sin'^^ -f cos'^^ = 1 ; 
(ii 1 tan ^ -i- (1 - cot ^) + cot ^ -^ (1 - tan A) = sec A cosec ^ + 1. 



REVISION EXAMPLES 135 

11. Draw the sine and cosine graphs, in the same figure, from 
^ = 10° to <9 = 20°. 

From the graph find the angle which satisfies 
sin ^ + cos ^= V2. 

12. Find an expression which will include all angles having 
a given tangent. Write down the values of tan 225^, tan 780", 
cot 1035% cot210\ 

Construct an angle, having given the cotangent. 
VS. Find a/cos A-^b/coi B + c/cos C in a form adapted to 
logarithmic calculation. 

14. In any triangle prove that (i) a = 6cos 6'+ ccos ^ ; (ii) 
«(6cos C-ccosi?) = ^'^-c^; (iii) rcoalA = as\n^Bsin\C. 

15. If the sides of a parallelogram are a, b, and the angle 
between them co, prove that the product of the diagonals is 

'^/a*-2a-b- cos CO + ¥. 

16. A vessel is steaming towards the East at 10 miles an hour. 
The beanng of a lighthouse as seen from the vessel is 42^24' North 
of East at noon, and 25° 12' East of North 25 minutes later. Find 
how far the vessel was from the lighthouse at noon, and find also 
at what time the bearing of the lighthouse will be due North. 

17. Assuming that a circle may be treated as a regular polygon 
with an infinite number of sides, show that the ratio of the 
circumference of a circle to its diameter is constant. 

What is the circular measure of the least angle whose sine is |, 
and what is the measure in degrees, &c., of the angle whose 
circular measure is '15708? 

18. Prove by a geometrical construction that 

cos 2^ = cos^^ — sin^x4. 
Solve the equation cos 2 A = (cos^ + sin Af. 

19. For what data will the solution of a triangle become 
ambiguous ? Explain this. 

Given J5 = 30°, c = 150, Z> = 50 ^/3, show that of the two 
triangles that satisfy the data one will be isosceles and the other 
right-angled. Find the third side in the greater of these triangles. 

Would the solution be ambiguous if ^ = 30°, c = 150, h = 75 ? 

20. AB is a horizontal line whose length is 400 yards; from 
a point in the line between A and B a balloon ascends vertically, 



136 KEVISION EXAMPLES 

and after a certain time its altitude is taken simultaneously from 
^ and jB ; at ^ it is observed to be 64° 15' ; at B 48° 20' ; find the 
height of the balloon. 

21. Find the radius of the circle circumscribing a triangle, in 
terms of its sides. If c^ = ci^ + W, show that this radius equals \ c. 

22. Define the trigonometrical ratios of A involved in the 
equation cot ^ + tan A = sec A cosec A ; and establish its tmth by 
a geometrical constiiiction. 

23. Prove that 



cos 



' ^/{a -x)-^{a-h) = sin ^ ^y{x -b)-r- {a - h) 



= cot""^ \/{a — x)-7-{x — b). 



24. Prove that sin 6 = tan 6-^ ^/l + tan2<9. 

Having given tan ^ = |, find sin 6, cos 9, and versin 0. 



25. If 6 is an acute angle whose sine is j%, calculate the value 
of tan ^ + sec ^. 

What would the value be if 6 were obtuse ? 

26. What is the angle between the diagonal of a cube and one 
of the edges at its extremity ? 

27. Obtain an expression for all the angles which have a given 
tangent. 

Find all the angles lying between —360° and +360° which 

satisfy the equation 

2 
tan'^x = tan re — 1 = 0. 

28. A circular wire of 3 inches radius is cut and then bent so as 
to lie along the circumference of a hoop whose radius is 4 feet. 
Find the angle which it subtends at the centre of the hoop. 

29. A triangle ABC has angle A = 34°, a = ll'O cm., c = 7*8 cm. 
Calculate the perpendicular from B on h, and the remaining 
angles and side of the triangle. 

30. In a triangle a = 14:, & = 37, c = 97 ; find the value of 
(i) acos^ + tcos^, (ii) cisinB — bBinA. 

31. If ABC be a triangle, and 6 an angle such that 

sin = 2 \/ah cos J C -f (a + h), 
find c in terms of a, h, and 0. 

If a = 11, h=.2ry, and C= 106° 15^', find c. 



REVISION EXAMPLES 137 

32. Find the area of a regular quindecagon inscribed in a circle 
of one foot radius. 



33. Find an expression for all angles having the same sine as 
the angle Oc. 

Solve the equation sin {0(. + x) + sin {3 + x) = 0. 

34:. An angle 0( is determined by the equations v^ = 2(/h, 
-b = tv sin a -Igf, tv cos 'X = a. Show that 

«2 tan^ Ot - 4 ha tan y + a'^-i hh = 0. 

35. Criticize the proposition that three measurements are suffi- 
cient and necessary to determine a triangle uniquely in shape 
and size. 

36. A square house, measuring 30 feet each way, has a roof 
sloping up from all four walls at 35° to the horizontal. Find the 
area of the roof. 

37. Draw up a table showing in three columns the values of 
10sin<9, 10 cos (9, and 8 sin ^ + 6 cos ^ for each 30° from 0° to 
360°. From the table draw, in the same figure, the graphs of 
?/ = 10sin^ and ?/ = 8 sin^ + 6cos^ ; and from the curves deter- 
mine approximately a value of 6 for which tan ^ = 3. 

38. Taking the earth as a sphere of radius 4000 miles, find the 
distance London travels in an hour in consequence of the rotation 
of the earth. (Latitude of London 51° 30' N.) 

39. ABCD is a quadrilateral in which AB and DC are parallel 
and 40 feet apart, and AB is 100 feet long. The angle DAB is 
72° 30', and the angle CBA is 38° 15'. Find the lengths of AD, 
DC, and CB, and the area of the quadrilateral. 

40. State the local time at the following places when it is noon 
at Greenwich. 

Cape Town 33° 56' S., 18° 25' E. Fiji 18° 0' S., 178° 0' E. 
Edinburgh 55°57'N., 3° 10' W. Singapore 1° 17' N., 103° 50' E. 



41. Define the cosine and the tangent of an angle, and show 
how to express the tangent in terms of the cosine. 

Having given that cos^l = '8, and that A is Jess than 90°, find 
the value of tan .4; and by means of the tables find the value 
of Af both from its cosine and from its tangent. 



138 REVISION EXAMPLES 

42. Prove that, in any triangle ABC, sin B: sin C = b:c. In 
the triangle ABC the angle CAB is 50°, the angle ABC is 65°, 
and the side BC is 4 inches long. Find the length of the side AB. 

43. Show how to find the height of a tree by means of a chain 
for measuring lengths and of an instrument for measuring angles. 

44 Find an expression for all the angles which have (i) a given 
tangent, (iij a given sine. 

45. Explain how it is that, tan ^ being given, tan 2^ is known ; 
but that, sin ^ being given, sin 2^ may have either of two values. 

46. Prove that the area of a triangle is \/.'i {s-a) {s- b) {s — c). 
Show also that the area is ^ c^ -f (cot J. + cot B . 

47. Find the radius of the circumscribing circle of the triangle 
for which A = 66° 30', B = IV 30', c = 200 feet. 

48. A ship is sailing due East at a uniform rate : a man on 
a lighthouse observes that it is due South at 1 p.m. and 16^30' 
East of South at 1.20 p.m. In what direction will he see it 
at 2 p.m.? 



CHAPTEE VIII 



PROJECTION. VECTOES 




Fi-. XLIV. 



67. If from the extremities of u line FQ, of definite 
length, perpendiculars PK, QL are let fall on a line AB, 
which may be produced if necessary, then KL is called the 
Projection of PQ on the line AB. 

Projections are subject to the same 
convention of sign as are abscissae 
and ordinates. Thus, in the above 
figure, KL is positive, but LK is 
negative. It follows that the projec- 
tion of PQ is not the same as the ^ 
projection of QP, so that the order 
of the letters in naming a line is 
of great importance when we are 

dealing with projection. When the direction of the line is to be 
taken into account as well as its length, it is called a directed 
length ; and we shall, in future, use the symbol {PQ) * to denote 
the directed length of the line from Pto Q. The number of unit> of 
length in that line we shall continue to denote by the symbol I'Q. 

Thus, in Fig. XLIY, the projection of (PQ) is (KL), 
and the projection of [QP) is {LK). 

Note. When we speak of the sum of directed lengths in the same 
straight line, tlie algebraical sum is always meant. Geometrically 
this means that we require the directed length between the starting- 
point and final point, and not the length of the actual path traversed. 

68. If the length of PQ is I, and if 6 is the angle 
between PQ and the line AB, then 

projection of {PQ) on AB = I cos 0. 



This is usually written PQ. 



140 PROJECTION. VECTORS 

Some care is necessary in applying this formula ; the safest 
plan is to keep ? and 6 both positive. 

Consider, for instance, the projection of {QP) in Fig. XLV. 

Imagine a line drawn from the initial point Q parallel to the 
line AB. Then it is seen that the angle between {QP) and AB 
is 6 + IT, while the length ^Pis I. 

Hence projection of {QP) on AB = lcos{d + 7r) = -I cos 6, 
Two other methods of treatment give the same result. 




A B 

Fig. XLV. 

In Fig. XLV the line QX is actually drawn parallel to AB ; but 
it is usually sufficient to imagine it. Then we may take the angle 
between (QP) and QX to be the negative angle XQP, i. e. 
— (n — 6); the length ^Pis positive so that 

projection of QP = lcos{ — Tr — 6) = — Zcos d. 

Or we may regard 6 as being the angle between {QP) and QX ; 
but this requires that the length of {QP) should be taken as —I, 
and so the projection of (QP) on AB = -/cos d. 

It will be found that, in all cases, I cos gives both the 
magnitude and sign of the projection of (PQ) on AB. 
Similarly, 

the projection of (PQ) on a line perpendicular to AB 

= I sin 6, 

69. Proposition A. The sum of the projections on any 



PROJECTION. VECTORS 



141 



Jhie of two sides {AB), (BC) of (a triangle is equal to the 
project ioti of the third side {AC). 





Fig. XLVI. 

In either of tbe above figures (or in any other figure) 
projection of (.45) + projection of (BC) = {ah) + {hc) 

= {ac) 
= projection of {AC). 

Proposition B. The sum of the projections on any line of 
the three sides {AB), {BC), {CA) of a triangle is zero. 

Sum of projections of iAB), iBC), {CA) = {ah) + (be) + (ca). 

Hence on the line of i^rojection we start at the point a and 
finish at the same point, so that the distance between the initial 
and final points is zero. That is, the sum of the projections 



Proposition C. In any closed figure ABC ... HK, the stm 

of the projectioyis of the sides {AB), {BC) ... {HE) equals the 
projection of(AK). 

Proposition D. In any closed figure the sum of the pro- 
jections of all the sides taJcen in order in the same direction 
is zero. 

Propositions C and D are proved exactly in the same way as Pro- 
positions A and B. 



142 PROJECTION. VECTORS 

Example. Prove that 

cos A + cos{120 + A)-\- cos (120 ~ ^) = 0. 




Fig. XLVII. 

Draw an equilateral triangle PQR, side a units. 
Draw a line OX inclined at an angle A to (QR). 
Then {RP) is inclined at .4 + 120 degrees to OX; and {PQ} is 
inclined at ^1 + 240 degrees. 

Project on OX; then, by Proposition B, 

a cos A + a cos {A + 120) + a cos (A + 240) = ; 
but cos (.4 + 240) = cos {360 -(120 -.4)} = cos (120-^); 
.-. cos A + cos (120 + A) + cos (120 -A) = 0. 



Examples VIII a. 

(These examples should be verified by drawing a figure to scale.) 

1. Show that the projection of a line on a line parallel to itself 
is equal to the projected line, and that the projection of a line on 
a line perpendicular to itself is zero. 

2. A line of length r, making an angle 6 with OX is projected 
on OX and at right angles to OX ; calculate the lengths of the 
projections in the following cases : 

(i) r = 5, ^ = 60°; (ii) r=-5, ^ = 120°; 

(iii) ;• = 5, ^ = 248° ; (iv) r = 5, d = 300° ; 

(v) r = -5, ^ = 330°. 

3. Two rods AB, BC, of lengths 5 feet and 10 feet respectively, 
are joined together at an angle of 135°. The rods are fixed in 



PROJECTION. VECTORS 143 

a vertical plane so that CB is inclined at 60'' to the horizontal, 
and the angle ABC is beneath the rods ; by projecting horizon- 
tally and vertically, find the inclination of the line ^C to the 
horizontal. 

4. By projecting a diagonal and two sides of a square on a line 
making an angle .4' with one of the sides, prove that 

cos (.4 + 45'') = (cos A - sin A) -r >/2. 

Find a similar value for sinu-1 + 45"). 

5. PQR is a triangle right-angled at Q, having the angle at P 
equal to A' ; FQ is inclined to OX at an angle B^. 

Prove by projection that 

cos {A + B) = cos A cos B — sin A sin B, 
and sin (A + B) = sin A cos B + cos A sin B. 



70. If the projections of a line on two lines at right angles are 
given, the length and direction of the projected line can be found, 
but not its actual position. 

Let r be the length of the line and the angle it makes with 
one of the lines of projection. Then r cos S and rsin 6 are known ; 
suppose these values are x and y respectively, so that >'cos^ = x 
and rsin 6 = ij. 



Then r = a/x- + i/- and tan 6 = y/x. 

The projected line has therefore a definite length and a definite 
direction ; it is the simplest example of a group of quantities 
called vector quantities or vectors. 

71. A quantity which possesses a direction as well as 
magnitude is called a vector. Yelocities and forces are 
examples of such quantities. The magnitude and direction 
can be represented by the length and direction of a directed 
straight line ; hence the properties of a directed straight 
line that depend only on its length and direction represent 
properties common to all vectors. 



144 PROJECTION. VECTORS 

72. Vector addition or Composition of Vectors. 

A displacement from A to B followed by a displacement from 
B to C produces the same result as a single displacement from 
^to a 




B 
Fig. XLVIir. 

Or we may regard the displacements as being simultaneous. 

Suppose a point to start from A and move along AB, and while 
this point is moving, suppose the line AB to move parallel to 
itself, the point B moving to C while the point travels from 
A to B. The result of the two simultaneous displacements is that 
the point has travelled from A to C. 

Hence the vector (AC) is called the resultant of the vectors 
(AB) and (BC). 

Finding one quantity equivalent to two or more of the same 
kind is equivalent to the process of addition in Arithmetic. 

If we use the sign + to denote this process, we have 
{AC) = {AB) + {BC). 

If P, Q, and R are the respective magnitudes of the vectors 
represented by {AB\ {BC), and {AC), and if 6 is the angle 
between the directions of {AB) and {BC) (in Fig. XL VIII the 
angle ABC is the supplement of 6) ; 

then 112 ^ P2 + Q2 + 2 PQ cos 6. 

Similarly, if a number of vectors are represented by the directed 
lengths {AB), {BC), {CD)...{HK), then their resultant is repre- 
sented by the directed length {AK). 

73. Besolution of vectors. 

In Fig. XLVIII the vector {AG) may be replaced by the 
two vectors (AB) and {BC). Viewed in this light they are 
called the components of the vector (AC). 



PROJECTION. VECTORS 



145 



When we talk of the component of a vector in a given direction, 
and no mention is made of the direction of the other component, 
it is understood that the other component is at right angles to the 

first. 

P 




Fig. XLIX. 

If (OP) in Fig. XLIX represents a vector of magnitude E inclined 
at an angle 6 to OX, then its projection {OX) represents the 
component along OX, and the projection (XP) represents the 
component perpendicular to OX. 

The vector is now said to be resolved along and perpendicular 
to OX. 

Resolving along OX, we find that the component is Rcosd. 

Resolving peqDendicular to OX, we find that the component is 
Psin^. 

74. All the work of § 69 on projections can be applied to 
vectors and their components. For instance, Proposition C gives 
the following proposition : 

The sum of the components of any number of vectors in a given 
direction is equal to the component of their resultant in that 
direction. 

Examples VIII b. 

[In the following examples the letters P, Q, R imply that the 
vectors are forces ; the letters u, v, w imply that the vectors are 
velocities. When possible, figures should be drawn to scale to 
check the calculation.] 

1. Find the resultant R in the following cases : 
(i) P= 17, <?= 13, e= 40"; 

fii) P= 17, (2= 13, 6^ 140'; 

(iii) P- 114. Q= 75, 6 = 65'; 

(iv) P= 123, (,^ = 496, e^nV. 



145 PROJECTION. VECTORS 

2. Find P when Q the other vector, 6 the angle between them, 
and R their resultant have the following values : 

(i) ^ = 176, i? = 249, e^ 72°; 

(ii) Q= 73, i^ = 193, 6= 110°; 

(iii) Q = 245, i? = 92, e= 130° ; 

(iv) Q= 36, 7? = 84, ^ = 20°. 

8. Show that, if the resultant of three forces is zero, the sum of 
their components in any direction is zero. 

4. Show that if three forces produce equilibrium (their resultant 
is, therefore, zero) they are parallel and proportional to the sides 
of a triangle. 

5. A boat is being rowed due E. at a speed of 6 miles an hour ; 
at the same time a current carries it due S. with a speed of 3 miles 
an hour ; find the magnitude and direction of the actual velocity. 

6. Find the resultant of velocities u and v inclined at an angle By 
when 

(i) 11 = 14, f = 16, 8= 180°; 

(ii) « = 14, i^ = 16, 0= 65°; 
(iii) u = U, v=lQ, e = 135°. 

7. Vectors of magnitudes 7, 8, 9 respectively are parallel to 
three consecutive sides of a regular hexagon. Find the sum of 
their components (i) parallel to, (iij perpendicular to, the middle 
one of these sides. Hence find the magnitude and direction 
of their resultant. 

8. Find the magnitude and direction of the resultant of four 
forces of magnitudes 5, 10, 15, 20 respectively, which act along 
the sides of a square. 

9. A stream flows at the rate of 2 miles an hour. In what 
direction must a man swim in order that he may actually go 
straight across the river, his rate of swimming being 3 miles 
an hour? 

10. A rod 5 feet long is hung by a string, attached to its two 
endS; over a smooth peg ; it rests, at an angle of 20° to the 
horizontal, so that the two portion^^ of the string are each inclined 
35" to the vertical. Find the length of the string. 



PKOJECTION. VECTORS 



147 



Projection on a Plane. 

75. If from every point in a line, atniight or curved, a perpen- 
dicular be let fall on a plane, the locuj; of the feet of the 
perpendiculars is called the projection of the line on the plane. 

If from every point in the boundary of a surface a perpendicular 
be let fall on a plane, the area bounded by the locus of the feet 
of the perpendiculars is called the projection of the surface on 
the plane. 

76. The angle between a straight line and its projection on 
a plane is called the angle between the straight line and the 
plane. It follows that the projection on a plane of a straight line 
of length I, making an angle a with the plane, is I cos (X. 

Any two planes, not parallel, intersect in a straight line. If 
from any point P in this line two perpendiculars FA, PB are 
drawn to it, one in each plane, then the angle APB measures the 
angle between the planes. 

77. If any plane surface, of area A, is projected on a 
plane making an angle a with its own plane ; then the area 
of the projection is A cos ol. 

*Step I. Consider a rectangle ABCD, having the side AB 
parallel to the plane of projection, and the side BC making an 
angle 0( with that plane ; then y is the angle between the plane 
of the rectangle and the plane of projection. 

Then, in Fig. L, abed is the projection of ABCD. 

D 




Fig. L. 
* A slight knowledge of solid geometry is assumed lu this proof. 

k2 



148 



PROJECTION. VECTORS 



Now Bb is perpendicular to the plane abed, and therefore to the 
line ab ; 

.-. Bb is perpendicular to AB ; 
but BC is perpendicular to AB ; 

.". AB is perpendicular to plane BCcb ; 
.•. ab is perpendicular to plane BCcb ; 
.-. ab is at right angles to he, 
i.e. abed is a rectangle. 

Hence art-a of abed = abxbc 

= ABxBC cos Oi 

= area of ABCD x cos CX. 

Step II. Consider a plane area with curved or rectilinear 
boundary. In the plane of the figure draw any line PQ parallel 
to the plane of projection. Then in the area we can inscribe 
a number of rectangles having the short sides parallel to PQ and 
the longer sides perpendicular to PQ. 




Fig. LI. 



The sum of these rectangles is less^than the original area, but may 
be made to differ from that area by as small a quantity as we please by 
making their width small enough ; and then the sum of their projec- 
tions will differ from tlie projection of the area by an even smaller 
quantity. Hence in the limit, when the width is indefinitely small, 
the sum of each set of rectangle;s will equal the area of the corre- 
sponding circumscribing figure. 

But the sum of projections of rectangles -- sum of rectangles x cos A ; 

.*. the area of projected figure =■ area of original figure x cos A. 



PROJECTION. VECTORS 149 

Examples VIII c. 

1. A pyramid VABCD has a square ba?e A BCD, side a, and the 
faces VAB, &c., are equilateral triangles. Find the length of the 
projection of VA on the base. 

Verify that the sura of the areas of the projections of the four 
faces is equal to a^. 

2. A square house, whose side is 28 feet long, has a roof sloping 
up from all four walls at 40° to the horizontal, find the area 
of the roof. 

3. Find, by projection, the area of the curved surface of a right 
circular cone, having height h, and semi-veitical angle 2 y. 

4. From a cone 6 feet high a smaller cone 2 feet high is cut off. 
If the radius of the base of the small cone is TG feet, find the area 
of the curved surface of the remainder of the large cone. 

Verify your answer by projecting this surface on the base. 

5. A circle with radius a is projected into an ellipse with semi- 
axes a and h ; show by projection that the area of the ellipse 
is TTCib. 

6. The vertical angle of a conical tent is 67", and the radius of 
the base is 5| feet; find (i) the slant height, (ii) the area of 
canvas used, (iii) the content of the tent. 

7. A pyramid on a square base is such that each of the other 
faces is an isosceles right-angled triangle, find by projection the 
angle between a triangular face and the base. 

Geometrical representation of imaginary quantities. 

78. In Fig. LII OA is of length r. 

By the usual convention a line OA drawn to the right is 

B 



A' A 

Fig. LII. 

considered positive, so that {OA) represents -fr. If now [OA) 
is turned through two right angles, it takes up the position {OA') 
and, by the usual convention, {OA') represents - r. Hence the 



150 



PROJECTION. VECTORS 



geometrical operation of turning through two right angles repre- 
sents the algebraical operation of multiplying by -1. Let us 
consider what the operation of turning through one right angle 
represents. 

This is an operation which, if performed twice in succession, 
turns through two right angles, which represents multiplica- 
tion by — 1. 

But the algebraical operation of multiplying by aZ-I, if 
performed twice in succession, multiplies by —1. 

Hence it seems reasonable that the operation of turning a vector 
line through a right angle represents the algebraical operation 
of multiplying by \/-l ; that is, (OB) at right angles to (OA) 
represents r x V^— 1, i.e. -%/ — Ir. 

In future we shall denote \/—l by /. 

79. With the interpretation of i suggested by the last section, 




cc + ii/ is represented by a vector line of length or followed by 
a vector line of length y at right angles to the first vector. 

X + iy ^ {0N)+ (NP) (Fig. LIII.) 

= (OP). (By vector addition, § 72.) 

Or, in words, or + iy is represented by the vector (OP), that is 



by a vector line of length >v/.r^ + f/^, 
with the positive direction. 



laking an angle tan"' 



80. For our purjioses this statement is more useful if reversed, 
VIZ. (OP) =^ x + iy 

= }'cos6+ ir Bin 6 
= (cos ^ + 1 sin B) r. 



PEOJECTIO^\ VECTORS 151 

Or, in words, the vector line of length r, in direction ^, repre- 
sents the magnitude r multiplied by cos ^ + /sin ^. This gives the 
important result that 

turning through an angle 6 represents multiplication by 

cos^ + j sin^. 
Hence 
turning twice in succession through d represents multipli- 
cation by cos ^ + /sin B repeated twice ; 
i.e. turning through 26 represents multiplication ly 

(cos + / sin 6f ; 
but turning through 2 d represents multiplication by 
(cos2^ + /sin2^). 
Hence the suggested interpretation of -y/— 1 or /, leads to the 
identity fcos 26+ i sin 2 ^) = (cos ^ + j sin 6)-. 

If this is verified by algebraic multiplication and by the use of 
the ordinary formulae for cos 2^ and sin 2^, it will be found 
correct. 

Carrying on the argument in the same way, we deduce that 
(cos 7id + i sin n 6) = (cos 6 + i sin 6)'\ 
where n is any positive integer. 

Again, turning through a half 6 is an operation which, if re- 
peated, turns through 6, and, therefore, represents a multiplication 
which, if repeated, multiplies by cos 6 + / ?in 6 ; 

• Le. (cos 16 + { sin 16)=> (cos 6 + i sin 6} '^. 

/ 6 6 \ - 

Similarly. ( cos - + / sin - j = ( cos ^ + / sin 6)" ; 

and ( cos - ^ + / sin - ^ ) = ( cos ^ -f / sin 6) i • 

a '1 / 

Lastly, turning through -6 cancels turning through 6, and, 

therefore, represents an operation which cancels multiplication 

by (cos (9+ i%\n6) ; 

i.e. {co8(-^) + /sin(-^)y = .cos^ + /sin^;-'. 

r!fimilarly, 

CCS { — n6) + i sin { — n6) = (cos 6 + / sin ^j~", 

where n is any positive quantity. 



152 PROJECTION. VECTORS 

We have now clediiced from the geometrical interpretation 
of \/-l. that 

(cosn^ + isinn^) = (cos ^ + i sin ^)" 
for all real values of n. 

This is known as De Moivre's Theorem. 

Example. Use De Mo iv re s Theorem to find \^\. 
cos 2 nTT = 1, sin 2 « 77 = 0, 
where n is zero or any integer. 

Hence 1 = cos2«7r-f /sin 2;?7r ; 

.'. ^T = (cos2n7r + isin2«7r)i 
= cosf ^?7r + isin§n7r. 

If >?-0, >v/r= cosO + /sinO = 1. 

If n = 1, ^T ^ cos Itt + / sin §77 = -^ ( - 1 + / yS). 

If n = 2, ^r=cos^7r + /sin^n- = H-1-?V3). 

If ;^ = 3, v^l = cos27r + » sin2n- =1. 

For other values of n it is seen that the three roots are repeated, 
Hence De Moivre's Theorem shows that there are three different 
cube roots of unity. They are, of course, the three roots of the 
equation ic^ — 1 = 0. The student should verify that the same 
roots are obtained hy Algebra. 

Examples VIII d. 

Represent graphically and by imaginary quantities the follow- 
ing vectors : 

1. (i) Magnitude, r = 25 ; direction 6 = y where a = tan-^g^^. 
(ii) „ r = 25; „ 6 = tt-OL; 

(iii) „ r = 25; „ 6 = tt + 0L] 

(iv) „ r=25; „ ^ = 27r-a; 

(v) ,. r = 25; „ 6=-0(. 

2. Show graphically that 

[x + iy') + {x" - iy") = (.r' + x") + i (//' 4- y"). 

3. Express the following in the form y (cos 6 -f / sin 6) : 

(i) 3+4/; (ii) 5 + 6/; (iii) 7-8/; 

(iv) -5-12/; (v) -5 + 12/; (vi) 8/. 

4. Interpret geometrically (cos a + /sinOK) (cos 0(-t sina) r; and 
justify your interpretation. 



PKOJECTION. VECTORS 153 

5. Show graphicall}^ that 

(cos 0( + -/ sin CX) (cos jS + / sin (3) = cos {0C + (3) + i sin {OC + /3). 

6. Verify De Moivre's Theorem by calculation, when n = 2, 3, 
-1, -2. 

7. Assuming De Moivre's Theorem, prove that v^ — l has three 
values, viz. -1, ^(1 + ^-3) and 1(1-7-3). 

8. («) Prove De Moivre's Theorem by induction when n is a posi- 
tive integer. 

(b) Deduce the proof when n is not a positive integer, by 
methods similar to those used for the Binomial Theorem in 
Algebra."^ 

Examples VIII. 

1. A man walks one kilometre in a direction 16 degrees North of 
East ; he then turns to the left, through an angle of 110 degrees, 
and walks one kilometre in the new direction. How far is he North 
and how far East of his starting-point ? 

2. Show that «cos^ + 6sin^ can be expressed in the form 
r cos {d - Oc). Illustrate by a figure. 

3. A number of rods are jointed together, and the two free ends 
are secured to two points A and B in the same horizontal line 
and distant c inches. If the length of the r^^ rod is «,. , and its 
inclination to the horizontal is 0,. (all the angles being measured in 
the same sense), prove that (i) 2 (ctr cos ^,.) = c ; (ii) 2 («,. sin 6,.) = 0. 
(See § 89, Example ii.) 

4. Prove by projection that 

sin(90 + ^) = cos^ and sin (270-^) = -cos ^. 

5. In what respects can a vector quantity be represented by 
a straight line? 

If three forces P, Q, R, acting at a point 0, are such that 
P/B\nQR= Q/smFP= R/sinPQ (where sin P^ denotes the sine 
of the angle between P and Q), show that the three forces produce 
equilibrium. 

6. A man walks one kilometre in a direction A"" North of 
East, one kilometre in a direction making 120" with the first 
direction, and one kilometre at an angle 240° with thefirst direction. 
Draw a figure showing that he has now returned to the starting- 

* See School Algebra, pp. 407, 435, 4G5. 



154 PROJECTION. VECTORS 

point ; and by considering the distances he has gone to the East 
and North write down two trigonometrical identities concerning 
the sines and cosines of A, 120 + -4, 240 + ^. 

7. Suggest a geometrical construction which may help to sum 
the series : 

(i) cos a + cos (a + /3) + cos (CX + 2 /3) + . . .n terms ; 
(ii) sin (X + sin {0( + ^)+ sin (a + 2^) + . . .w terms. 
Deduce that both these sums become zero if «,3 = 2it. 

8. A body which weighs 12 lb. is kept at rest by means of two 
cords, one being horizontal and the other inclined to the horizontal 
at an angle whose tangent is | ; find the forces exerted by the cords. 

9. A mine shaft is 1650 feet in length. It slopes downwards at 
an angle of 45° to the horizon for a certain part of its total 
length, say x feet, and at an angle of 35° for the rest of its length. 
If the total depth reached is 1000 feet, obtain an equation for x, 
and hence calculate x. 

10. A man playing five holes of a golf course first walks 260 yards 
due East, then 140 yards 20° South of East, then 300 yards due 
South, then 200 yards 40° West of North, then 220 yards 30° West 
of South, thus arriving at the fifth hole. Find how far the fifth 
hole is from the first tee. 

11. The perpendicular from the origin on a straight line equals 
2) and makes an angle a with the axis o( x; by projecting the 
co-ordinates of any point on the line show that the equation of 
the straight line may be pat in the form a; cos a + 1/ sin Oc = p. 

(This is known as the perpendicular form of the equation of 
a straight line.) 

Hence find the length of the perpendiculars from the origin on 
the lines whose equations are (i) Sx + 4ij = 7 ; (ii) 5x-l2ij = 2; 
(iii) x + 2t/ = 6. Verify by drawing to scale. 

12. The co-ordinates of a point referred to rectangular axis 
OX, OY are x, y; referred to two rectangular axes OF, OW 
through the same point the co-ordinates are |, //. Prove by 
projection that ^ = xco^oc + ijs'm OC, where Oi is the angle between 
OX and or. 

Find three other similar relations connecting ^, »/, x, //, 



CHAPTER IX 

FORMULAE FOR (i) THE FUNCTIONS OF THE 
SUM OR DIFFERENCE OF TWO ANGLES, 
(ii) THE SUM OR DIFFERENCE OF THE 
FUNCTIONS OF TWO ANGLES, (iii) THE FUNC- 
TIONS OF THE DOUBLE ANGLE AND THE 
HALF-ANGLE 

81. To express cos (A + B) in terms of the sines and 
cosines of A and B. 





Fig. LV. 



Let OX be the initial line ; and let the revolving line first turn, 
through the angle A to the position OA and then continue to 
turn through an additional angle B to the position OB. Then OB is 
the bounding line of the angle A + B. Along OB measure a length 
OP=r units. 

Project OP on the initial line, produced backwards if necessary 
also project OP on OA, produced backwards if necessary. 

Figures LIV and LV show two of the many possible cases. 



156 FORMULAE FOR FUNCTIONS OF THE SUM 

In all cases 
the projection of (OP) on OX 

= sum of the projections of (OK) and (KP) on OX ; 
i.e. rcos{A + B) = (OK) cos A + {KP) cos (^ + 90) 
= r cos B cos A + r sin B { — sin A). 
. Hence cos (A + B) = cos ^ cos £ — sin A sin B. * 

Several proofs of this have already been given, but the earlier 
proofs have implied that A and B are together less than two right 
angles ; this proof applies whatever be the values of A and B. 

Exercises, (i) Deduce the formula for sin (A + B) by substitu- 
ting 90 -v4 in place of A, and -B m place of B. 

(ii) By similar substitutions deduce the formulae for cos {A — B) 
and sin(.4 — J5). 

(iii) By projecting perpendicular to OX, find the expanded 
form of sin {A -f B). 

(iv) Modify the construction so as to prove directly, by pro- 
jection, the formula for sin (A-B) and cos (A~B). 

(v) Complete the following formulae : 
cos {A -¥B) = 
cosU-5) = 
nn{A + B) = 
sin(^-^) = 

(vi) Learn these formulae in woids, as : 

cos sum = cos cos - sin sin. 



* When Fig. LV is being used it must be recollected that {OK) is 
negative, and tliat its inclination to OX is XOA not A'OK, see § 68, 
If (07v) is regarded as positive, its actual length is — r cos B ; but the 
angle is tlien JCOK, the cosine of wliich is —cos A. Whatever way it 
is taken, the projection of {OK) on OA' is found to be r cos £ cos A. 



OK DIFFEEENCE OF TWO ANGLES 



15' 



82. The followinrr proof does not involve any knowledge of 
projection ; its chief drawback is that it applies only to the case 
when A + B is less than a right angle. It is easily modified to 
suit any other given case. 




Let XOA = A and AOB ^ B ; 
then XOB = A + B, and in Fig. L VI is less than 90°. 

Take a point P in the bounding line oi A + B; 
let fall PN perpendicular to OX-, 
PK ,. „ OA; 

„ KL ,, ,, PX and. therefore, parallel to OX 

KM „ „ OX. 

ON OM-MN 



goh{A+B) = 



OP 


OP 


OM 
OP 


LK -\ 
OPj 


OM OK 
OK OP 


LKPK 
PK OP 



The spaces in the bracketed line (which does not appear in the 
completed work) are filled in with the hypotenuses of the triangles in 
which the respective numerators occur. 

Now angle LPK = 90° - LKP = LKO = A ; therefore 
LK PK = sin A. 

Hence cos (A + B) = cos A cos B - sin A sin B. 

From the same figure, prove that 
s'm(A + B) = 

To find the functions of A—B, the angle AOB must be made on 
the negative side of OA. The point P must be taken in the 



158 FORMULAE FOR FUNCTIONS OF THE SUM 

Dounding line of A — B, and the construction and proof proceed as 
before. It is found that 

cos{A-B} = 

sin (.4 -5) = 

Exercises, (i) Prove the four formulae when A and B are each 
less than 90^ but ui + B is greater than 90^. 

(Make the same construction as when ^ + i>' is less than 90^, and 
pay careful attention to the signs of the lines.) 

(ii) Prove the four formulae when A and B are each obtuse 
and together greater than 270°. 

Examples IX a. 

1. By using the formulae of § 81, verify that sin (90 — -4) = cos^ 
cos(90 + ^)= -sin^, sin (180-^) = sin .4, cos (180-^1 j= -cos^ 
sin(270 + ^)= - cos ^, cos (360-^) = cos^. 

2. Express cosTO"" in terms of the functions of (i) 40° and SO"* 
(ii) 45° and 25° ; (iii) 95° and 15° ; (iv) 35°. 

3. Express sin 40° in terms of (i) 30° and 10° ; (ii) 25° and 15° 
(iii) 70° and 30° ; (iv) 20°. 

4. From the expansions of sin(^ + i^) and coii{A + B) deduce 
the expansion of tan(^ + J5) in terms of tan^ and tani?. 

5. From the expansions of sin(^ — ^) and cos (^-^) deduce 
the expansion of tan {A - B) in terms of tan A and tan B. 

6. Verify that sinO° = and cosO° = 1 by using the formulae 
for^-^. 

7. Show that (i) sin (^ + -B) cos -B - cos (^ + 5) sin 5 = sin 5 ; 

(ii) cos {A + B) cos B + sin {A + B) sin B = cos B. 

8. From the formulae for A + B deduce that 

sin 2^ = 2 sin -4 cos ^ and cos2^ = cos^vl — sin'-^. 
What is the value of tan 2^ ? 

9. Find the values of 

(i) sin(^ + jB)-} sin(^-^) ; (ii) cos ( J + B) -f cos (.4 - £) ; 
(iii) sin {A 4- B) - sin {A - B) ; (iv) cos {A + B)- cos {A - B). 
Account for the signs of (iii) and (iv) from first principles. 



OR DIFFERENCE OF TWO ANGLES 159 



10. Prove thiit 

sin A = sin }, (A + B) cos \ {A-B) + cos \ [A + B) sin 1{A- B). 
Prove similar results for cos A, sin B, and cos B. 

11. From the results of 10 deduce that 

sin ^ + sin J5 = 2 sin \{A + B) cos \{A- B), 
and three similar results. 

12. Prove that 

(i) cos- 6 -f cos- (p — 2 cos 6 cos (p cos ($ + (p) = sin- (d + cf)); 
(ii) sin- 6 + cos- — 2 sin ^ cos sin (^ + </>) = cos^ {B + </>). 

83. Sums and difi'erences of sines or cosines ex- 
pressed as products. 

These formulae are most easily derived from the formulae of 
§ 81, as suggested in Examples IX «. They can be proved inde- 
l)endently by projection. 

Make the angle XOA = .1 and the angle XOB = B.* 
On OA and OB take lengths OP, OQ respectively, each equal to 
r units. Join PQ. 

B 





Fig. LVII. Fig. LVIII. 

Bisect the angle QOP by a line cutting PQ at R. 

Then the angles BOP, ROQ each equal 1{A-B) ; and the angle 
XOP = l{A + B). 

From congruent triangles EP = RQ, and PRO is a right angle. 

The projection of (OP) = sum of projections of (OR) and (RP), 
and the projection of (0^; = sum of projections of (02?) and (RQ). 

* Notice the difference between tbi> construction and tbc construc- 
tion of § 81. 



160 FORMULAE FOR SUM OR DIFFERENCE OF 

.•. projection of ( OP j + projection of (0^)^*2'^ projection of 
(OR), since projections of {HP) and (BQ) are equal but opposite. 
Projecting on a line perpendicular to OX, we have 

r sin A + r sin B = 2'^'' the projection of {OB). 
But {OR) is the projection of {OP) on the direction OR, 
i.e. {OR) = rco&\{A-B). 

rsin^ + rsin J5 = 2rco&\{A-B)^m\{A + B). 
Hence sin ^ + sin £ = 2 sin ^ (^ + B) cos \{A-B). 
From the same figure, by projection on OX, we have 
cos^ + cosjB = 2cos|(^ + jB)cos^(^-jB). 
Again, projection of (OP) -projection of (0$) = 2^^ projection 
of {RP). 

Hence sin^ — sin P = 2 cos | (A + P) sin \{A — B), 

cos^-cosP= -2sin|(^-P)sin J(^ + P). 
The proofs apply to all cases whatever be the magnitudes of 
A and P. 

The reason for the negative sign in the last of these formulae is 
obvious, for, if A> B, then cos A < cos P. 

It is useful to learn the formulae in words, it being understood 
in all cases that the greater angle is put first. 

sine + sine = 2 sine half sum cos half difference. 



84. Products of sines and cosines expressed as sums 
or differences. 

In the formulae of the last section put 

i(^ + P) = X, \{A-B)=Y', 
so that A^X+Y, B = X-Y. 

Then sin(X+ r) + sin {X- Y) = 2 sin X cos Y, 

i. e. 2 sin X cos Y = sin {X+ F) + sin(X- Y). 

Similar results are obtained from the other formulae. If we 
replace X by ^ and Y by P, the formulae become 
2 sin ^ cos P = sin (.1 + P) + sin (.1 - P), 
2 cos .1 sin B = sin (.1 + P) - sin (.1 - P), 
2 cos.l cos P = cos {A + P) -f cos {A - B), 
2 sin A sin P ^ cos (.1 - P) - cos {A + P). 
These are more easily proved direct from the A-vB and A—B 
formulae. 



FUNCTIONS OF TWO ANGLES 161 

In using these formulae it is usual (but not necessary) to put the 
greater angle first ; this shows why there are distinct formulae for 
2 sin .4 cos i? and 2 cos .4 sin i?. 

Express the four formulae in words : 

Twice sine cos = sin sum + sin difference. 



Examples IX b. 

1. Apply the formulae of §§ 83, 84 to the following cases and 
verify from the tables : 

(i) A = 70°, B = 30° ; (iii) A = 72°, 5 = 18°; 

(ii) A = 110°, 5 = 75° ; (iv) A = 78°, B = 46°. 

2. Prove, from the formula for sin A + sin B, that 

sin 2 ^ = 2 sin 6 cos d, 
and, in a similar way, show that 

1 + cos ^ = 2 cosH ^, 1 - cos JL = 2 sinH ^. 

3. Prove that 

(i) sin ^ + cos ^ = ^/2 cos {A — 45) ; 

(n) sm A — cosB= - 2 sm ( 45 7,— 1 sm ( 4o ^ — j ; 

/•••x . • -^ ^ /.K A-B\ { ,^ A + B 
(ill) cos A + smB = 2 cos f 45 ^ — 1 cos ( 45 + 

4. Prove that (i) sin 50° + sin 130° = 2 cos 40° ; 

(ii) cos 50° - cos 130° = 2 sin 40°. 
Verify these by squaring and adding. 

5. Prove that (i) 2 cos 40° sin 50° = 1 - sin 10° ; 

(ii) 2 cos 40° sin 40° = sin 80°; 
(iii) 2 sin 64° sin 26° = cos 38°. 
Verify this last result from the tables. 

6. Fill iu the right-hand side of the following : 
(i) sin /0° + sin50° = (ii) cos 30° -cos 110° = 

(iii) 2 sin 75° cos 10° = (iv) sin 37° + cos 24° = 

(v) 2 cos 84° cos 72° = (vi) cos 79° -cos 52° = 

(vii) sin 75° -sin 116°= (viii) 2 cos 80° cos 35° = 

(ix) cos 24° -sin 76° = (x) 2 sin 17° sin 48° = 

1216 L 



162 FORMULAE FOR FUNCTIONS OF 

(xi) 2 cos 73° sin 15° = (xii) 2cos U°cos 166° = 

(xiii) cos |7r-cos|rr = (xiv) sin 1 7r + COS J tt = 

(xv) 2 sin 43° cos 47° = (xvi) 2 cos 97° sin 46° = 

(xvii) sin 81° + sin 10° = (xviii) sin 49° -sin 53° = 

(xix) 2sin79°sinl5° = (xx) cos 43° -cos 216° = 

7. Prove that 4 cos (75° + A) sin (75° - ^) = 1 - 2 sin 2 ^. 

Formulae for the double angle and half- angle. 

85. It lias already been shown in § 51 that 

sin 2 ^ = 2 sin A cos A; 
cos 2A = co&'^A - sin^^ 
= l-2sin2^ 
= 2jiQsM-l. , 
The proof there given assumed that 2 J. is less than 180°. 
If we put A instead of B in the A + B formulae the same results 
are obtained ; thus they are true for all values of ^. 

The results can easily be proved independently by projection ; 
the proofs are the same as in § 81, A taking the place of B. 

86. From the last section, by putting -^^ in place of A, we 
have cos'* ^A — sin'^ -| J. = cos A ; 

also cos'^^^ + sin^^ J. = 1. 

Add 2cos^i^ = l + cosX 

Subtract 2 sin^ ^A = 1- cos A. 

Hence cos^^ = + Kv^l + cos^) ; 

sin|^= +|(-v/l -cos^). 
If the value of A is given, there is no ambiguity of sign. If, for 
instance, A = 140°, then ^A = 70°, and the sine and cosine are 
both positive; if ^ = 264°, then -^.4 = 132°, and the sine is 
positive, the cosine negative. 

If the value of cos^ is given but not the value of A, the 

ambiguity cannot be removed. Suppose cos .4 = ^, then A may 

have any value of the form 360° n ± 60°. Hence I A may have any 

value given by 180° « + 30°. If we tabulate these values, we have 

r igle cosine sine 

30° +|-v/3 +^ 

150° -^^/S +1 

210° -},^/3 -I 

330° +^^3 -I 



DOUBLE ANGLE AND HALF-ANGLE 163 

87. Tangent formulae. 

From the sine and cosine formulae the following tangent formulae 

are derived ; the proof of the first only is given : 

, , „, sinU+5) 

tan \A-{-B) = — — =r( 

' cos (^ + 5) 

sin A cos B + cos A sin B 

cos A cos B — sin A sin B 

_ ^n A + tan B (By dividing throughout by 

~l-tanAtanB cos^eo3J5.) 

Similarly, 

tan (^-5) = 

tan 2.^ = 



1 A /I -COS A 

tan I A = + A z 

^ - A' 1 + cos A 

1-cosA sin A ,„ ,. ... , 

= + — -. — - — or + _ . (By rationaliznig. ) 

- sin A -1 + cosA ^ ^ °' 

88. These may all be proved directly from the figures used for 
the sine and cosine formulae ; e.sr. in Fier. LVI. 



tan (^ + 5) = '^, 



MK^LP 



OM- 
MK 
OM 


LK 

LP 
"^ OM 



(By dividing so as to make the first 
~ LK PK' term in the denominator to be L) 
PKOJl 
The triangles LPK, OKM are similar ; 
LP PK , ^ 

TT L / i T,\ tan A + tan B 

Hence tan(^ + ^)=:, — 7 tt ^• 

1-tan^tan^ 

Exercises. Prove that 

tanyl + tan5 + tan(7 — tan^tan^tanC 



(i) tan(^4-5+C) = 



l-tanjBtanC-tanCtan^-tan^tan^ 



(ii) tan.i + ta„£ = '^^^^^'; 
cosJ-cos-B 



(iii) tan X- tan 5 



sin(^--B) 

cos^ cos5 

l2 



164 FORMULAE FOR FUNCTIONS OF 

(iv) cot A + cot B = -. — -—. — =, ; 
sill A sm B 

{,)cotA-ootB =-^i'^Mz^'. 
sm A sm A 

Example. To prove that tan'^ g\ + tan ~ ^ ^ J y = tan~ ' ^^q • 

Let A = tan-^ ^g, B = tan-^ ^tf ; 

so that tan A = ^^, tan B — o|y . 

tan A + tan B 



tan (^ + 5) = 



1 — tan A tan ^ 



■9^ + - 



1 ]L V _1- 

■■■ 99 "^ 2'59 



239 + 99 338 



23900-240 23660 



i. e. tan ^ t^^^ + tan"^ 239 = tan" 



-1 1 



7(J' 



Examples IX c. 

1. Prove that (sin A + cos Af = 1 + sin 2^, 
a nd (sin A - cos Af = 1 - sin 2 ^. 

2. Assuming the values of sin 45°, cos 45°, tan 45°, deduce 
sin 90°, cos 90°, tan 90°. 

3. Find a formula for cot 2^ in terms of cot^. 

4. Show that sin ^A = 3sin^ -4 sin^ JL. Explain how it is that 
there are three values of sin J. when sin 3 J. is given. 

5. Find the values of tan 22^°, tan OTf, tan 157i°. 

6. Prove that 2sin-|^ = + -y/l + sin A ± ^/l-smA. 
Find sin ^.4 when sin^ = |. Illustrate by a figure. 

7. Find cosl^ when sin^ = ^. Illustrate by a figure. 

8. Prove that (i) sin 2^ = 2 tan^^(l +tan2^) ; 

(ii) cos2^ = (l-tanM)~(l+tanM); 
(iii) tan 2^ = 2 tan ^ ^ (1 - tanM). 

9. Prove that sin ^A = 3 sin^l — 4 sin^^ ; 

cos 3^ = 4cos^u4-3cos^ ; 

tan 3^ = (3 tan^- tan^^) ^(1 - 3 tanM). 



DOUBLE ANGLE AND HALF-ANGLE 165 

10. Show that 

(cos A + sin Af + (cos A - sin Af = 3 cos ^ — cos 3 A. 
11 (rt). Show that sin^^ + cos^^ = + \/l + sin^, 
and sin ^A — cos ^ J. = + v^l — sin A. 

(b) Having given 4 sin 54°= ^/b + l, apply the formulae in 
(a) to find sin 27° and cos 27°, explaining how the ambiguities of 
sign are cleared up. 

(c) Show that 8(sinU2°-cos2 78°) = -/S + l. 

12. Prove that 

fA R r — ^^'^^^tan^ + tan C— tan^tan5tan C 
'^ ~ 1-tan^tan C — tan C7tan^— tan^tan^ 
Deduce the formula for tan dA- 
What can be deduced ifA + B + C equals (i) 180°, (ii) 90° ? 

13. If tan ^ = If and tan B = 2%, show that 

A + B = {4:n + l)l7r. 

14. Show that 

cos ^ + cos 3 ^ + cos 5 ^ + cos 7 ^ = 4 cos ^ cos 2 ^ cos 4 S. 

15. Find all the solutions of the equation 

sin 6 sin 3 ^ = sin 5 6 sin 7 6. 

16. If tan^ = I, tan 5 = §, tan C = fj, and each angle is acute, 
prove that A + B + C = ^7r, 

17. If tan^ = tanjcxtani^, show that 

tan 2 ^ = (sin OC sin ^) -r (cos y. + cos /S). 

18. (i) If ^ = tan-4, find tan 2^. 

(ii) Show that 2 tan"^ | + tan-^ } = in. 

19. Prove that 

cos2^-cos2^ = 2(cos-^-cos2 5) = 2(sin-5-sin^^). 

20. Prove that 

.. _J sec4^ 

^^^ a + bcosd~ {a + b) + {a-b)ta.n'^^B'' 

1 1+tan^l^ ^ 

^"^ acos^ + fesin^ ~ a + 26tan|^-atan2i^ * 

21. Solve the equations 

(i) x2-^/2sin(i7r + Oi)a; + isin20( = 0; 
(ii) «2_2cot2/3.a;-l = 0.'' 

22. Solve for OC and Fthe following equations : 

2ag = V- sin 2CX, 2bg = V^ sin^ y. 



166 FORMULAE FOR FUNCTIONS OF 

23. A hemispherical shell of radius 16 inches rests with its rim 
on a horizontal table ; a rod is hinged to a vertical wall, 25 inches 
from the centre of the shell, at a point 5 feet above the table. 
The rod is in the same vertical plane as the hinge and centre of 
the shell, and touches the shell. Find its inclination to the 
vertical. 

Oral Examples. 

(a) (i) sin(P-^) = (ii) cosX + cos r = 

(iii) cos (90- 1 J+^) = (iv) sin 270° 

(v) 2 sin a cos /3 = (vi) cos^ 6 - sin^ 6 = 

(vii) tan (A-B) = (viii) sin ^ - sin C = 

(ix) cos245°-sin2 45°= (x) cos 2.1 = 

(6) (i) cos(C+.4) = (ii) sini? + sinC 

(iii).2sini(5-f Cji(P-C)= (iv) cos^ + cos(/) = 

(v) 2cos2 1(7-1 = (vi) tan25 

(vii) sin (180-5 + C) = (viii) cos^TS^ + sin^TS^ = 

(ix) cos{^A + B + lA-B)= (x) sin (360-2 Cj = 

(c) (i) sin 2 5 = (ii)sin(P+^) 

(iii) cosin-oc + li) = (iv) 2siniCcosJC = 

(v) 1-2 sin^ B = (vi) cos^ | C- sin^ i C = 

(vii) cos2 1 C + sin^ A C = (viii) tan (B-C) = 

(ix) cosC-cos^ = (x) (sin j5 + cos 5)^ = 

(d) (i) sin^cosC-sin^sin C = (ii) cos(X-r) = 
(iii) sin 3^ = (iv) sin-^i? = 

(v) cos 5 + cos (7 = (vi) 2 cos 5 cos C = 

(vii) sin^ {B + C) + cos- {B + C) = (viii) 2 cos- ^ ^ - 1 = 
(ix) (cos 1^- sin 1 .1)2 = (x) tan (90 -C) = 

89. The preceding formulae load to a number of useful identities 
in the cases where A + B + C=90° or 180". The method of dealing 
with these is shown in the following illustrative examples. 



DOUBLE ANGLE AND HALF-ANGLE 167 

Example i. In any triangle tan H^"" ^) = r — ^^^ \ ^- 

[Here - — - gives the clue to the proof.] 

+ c 

b c 

By the sine formula, -. — ^ = ~ — ;;,' 

*' ' sm j5 sm C 

sin B b 
I.e. . ^ = -, 

sm C c 

^ , i T -i T sin 5 — sin b — c 
Componendo et dividendo, -; — ^^ -. — i^ = , ; 

2cosl(B+C)sm^iB-C) _ b-c ^ 
28in^{B+C)cos^{B-C)~ b + c' 

tan^(g-CO _ b-c ^ 
^•®* tani(.B+(7j ~ b + c' 

hut A + B+C=180''; .'. IA + 1{B+ C) = 90. 

Hence tan |(5+ C) = tan (90 -M) = cotj^. 

b — c 
Substituting above, tan ^(-B - C) = cot ^A. 

This formula has been proved geometrically in § 53 ; it is usually 
proved by the method given above. 

Example ii. In any triangle 

^^~~cUS^-\-€o^^ + cosC = 1 + 4 sin i A sin J B sin i C. 

L.H.S. = cos J. + cos^ + cos C 

= 2 cos * (^ + jB) cos 1 U - -B) + 1 - 2 sin^ -i C 

= 2sin|Ccosi(^-5) + l-2sin|Ccos^(^ + J5), 

since ^C = %-l{A + B) 
= l + 2sin|C(cosi^-^-cos^^ + ^) 
= l+4sin|^sin^5sin"?C. 

The symbol 2 cos A is sometimes used to denote 

cos^ + cos5 + cos C; 

and nsin^ to denote sin ^ sin 5 sin C. The above result can be 
written: 2 cos^ = 1 4-4nsin^^. 



168 VARIOUS ILLUSTRATIVE EXAMPLES 

Example iii. In any triangle ^cos'^A = l — 2Yicos A. 

(Questions involving the sum of the squares of sines or cosines are 

usually solved by expressing these squares in terms of the cosine of 

the double angle.) 

2 2 cosM = 2 cos2 A + 2 cos'^B + 2 cos^ C 

= 1 + cos 2^ + 1 + cos 25 + 2 cos^ C. (Note that one angle 

is left unchanged.) 
.-. 2 cos^^ = 1 + cos (^ + B) cos (A-B) + cos^ C 

= 1 - cos C cos (^-5) -cos Ccos (A + B), 

since C= 180 -(^ + 5) 
= 1 - cos C [cos {A ~ B)-cos(A + B)] 
= 1—2 cos Acqs B cos C. 

Example iv. Solve the eqimtion 

sin + sin 2e + sinS0 + sin 4:6 = 0. 
Rearrange sin ^ + sin 4 ^ + sin 2 ^ + sin 3 ^ = 0. 
Use formula for sum of two sines 

2sin#^cos|^ + 2sin|^cos|^ = 
.*. either sin f ^ = or cosp+cos^^ = 
i.e. -^6 = nrr or 2 cos^cos ^^ = : 

i.e. 6 = ^nrr, or cos ^ = or cos^^ = 

i.e. ^=(2n + l)*7r or |^ = (2/i4- l)i7r. 

Hence the complete solution is 

^=(2w + l)7r, (2n + l)i7r or fnTT. 

Example v. To prove that 

r = 4:B sin J A sin I B sin i C. 

From the figure of § 59, 

rcot^5 + rcot^C= a, 

/coshB cosiCX ^ „ . , 

r . l^ + -7-^-y, ) = 2i? sin ./, 

\sm|5 sin|C/ ' 

siniBcos^C+cos^^sin-JC ^^ . 

I.e. r —^ . ', p . , V, ^ = 27?sm^, 

sinf5sin|(7 ' 

sini(J5 + C) ^^ . , 
I.e. r . {1. . '^ = 27?sin^, 

sm|5sin^C 

i. e. r _._;o^i;^ = 4 7^ sin I A cos I A ; 

sinl^sin^C 

''sincei(P+C) = 90-|^; 

.-. >• = 4 7? sin I ^ sin ^5 sin J C. 

Exercise. Prove that r^ = 4 7i? sin J ^ cos ^ 5 cos ^ C. 



I.e. 



VAKIOUS ILLUSTEATIVE EXAMPLES 169 

Example vi. To show that the distance between the circum- 
centre and in-centre = \^{Ii- — 2Br). 

A 




In Fig. LIX, with the usual notation, 

BD = E sin A, DO = R cos A, 
BP= 1' cot IB, PI=r. 
Or- = {BP-BDf + {IP-ODf 

= {rcoi\B-R%\nAf + {i'-Rco^Af 

= i?2_2i;-(sin^coti5 + cos^) + »^(l + cot45) 



sin|^ sin^l^ 

^0 nr. sinU+*5) ,^ sini^sinJ^sinlO (Substitut- 

smi^ sm'^i^ mgforr.) 

^o or> cos^{A-C)-2smlAsm^C 

= U' — 6iir -. — , , 

sin ^5 

since A + },B = ^0 + lA-lC 

siniB 
= R!^-2Rr. since i(^ + Cj = 90 -i5. 

This is more shortly proved by Pure Geometry ; but the method 
used here is a general method to find the lengths of lines connected 
with the triangle. 

Example vii. To p'ove that 

sin A + sin {A-\-B)-\- sin {A + 2B)-{- ... to n terms 



sin h nB sin {A + ^n—l B) 
sin I B 



170 VARIOUS ILLUSTRATIVE EXAMPLES 

Let S* = sin ^ + sin (^ + i?) + . . . + sin (A + n - 1 B). 
Multiply by 2sinjP. 
Then 2smlB.S 
= 2sin^sini^ + 2sin(^ + ^)sini5+ ... +2sin(^ + n-l^)sini5. 
Use the formula for the product of two sines. 
2sini^.5'= cos(^-ij5)-cos(^+|5) 
+ cos (A + IB)- cos [A + IB) 
+ .... - 



+ cos (.4 + i2n- 35) -cos (^+12^1-1^) 
= cos(^-i5)-cos(^ + i2M-15); 
^^m\)iB%m{A + \n-lB) 
sin^?i5 

Note, Compare tliis with the formula for the sum of n terms of an 
Arithmetic Progression. Notice that A + \n — \ B = half the sum of 
the first angle {A) and the last angle (^4-n — 1 B), 

Examples IX. 

1. Prove the following identities : 

(i) sin 3 ^ = 4 sin A sin (60^ + A) sin (60'' - A) ; 

(ii) sin 3 A sin^^ + cos ZA cos^ A = cos' 2 A ; 
(iii) (l-2sin2^)-f(l + sin2^) = (1 -tan^) f (1 + tan^) ; 

,. . tan (45° + ^)+ tan (45°-^) ^ , 

(IV) r~ y-Ar^o-^Tx , 7TE5 7\ = COSeC 2 ^ ; 

^ ' tan (45° + ^) -tan (45°-^) 
(v) sin {y->iZ-x) + sin {z + x-y) + dn{x + y-z)-^\n{x + y + z) 

— 4 sin a: sin y sine'; 
(vi) cot J^ — cot^ = cosec^ + cosec-|^; 

(vii) cos 4 ^ + 2 (cos ^ + sin J.)* = 3 + 4 sin 2 A ; 
(viii) sin ^ + sin 5 = sin (^ + -B) + 4 sin \ AQm\B sin \{A + B)', 
(ix) sin ^ - 3 sin 3 ^ + 3 sin 5 ^ - sin 7 ^ = 8 sinM cos 4 ^ ; 
(x) cos(^ + J?+C) 

= cos A cos B cos C — cos A sin 5 sin C — sin A cos B sin C 

— sin^ sin 5 cos C\ 
(xi) cos \ A (2 sin A - sin 2 A) = sin^ I A (2 sin ^ + sin 2 ^) ; 

(xii) cos A-vco%B + cos C + cos (^ + i? + C) 

= 4 cos ^ {A + B) cos I {B + C) cos },{C-\- A). 



SUM AND DIFFERENCE FORMULAE 171 

2. If A, B, C be the angles of a triangle, show that 
(i) tan A + tan B + tan C = tan A . tan B . tan C ; 

(ii) sin 2 ^ + sin 2 5 + sin 2 C = 4 sin ^ sin 5 sin C ; 
(iii) sinH^ + sin2i5 + sin2iC + 2sin*^sin*5sin^ C= 1; 
(iv) sin ^ + sin jB + sin C = 4 cos \ A cos \ B cos \ C ; 
(v) cot A cot 5 + cot A cot C + cot 5 cot C = 1 : 
(vi) cot A + cot B + cot C 

= cot ^ cot B cot C + cosec .4 cosec B cosec C ; 
(vii) tan B tan C + tan Ctan J. + tan ^ tan B 

= 1 + sec A sec 5 sec C ; 
(viii) cos ^ sin (5 -C'j + cos 5 sin (6'-^j + cos Csin(^-5) = ; 
(ix) (tan A + tan B) (tan A - cot C) = sec^^ ; 
(x) tani5tan^C + taniCtanM + tani^tani5 = 1. 

3. Show geometrically that sin(^ + 5) = sin^cos^ + cos^sini? 
when each of the angles A and B is between \tt and it, and A + B 
is less than frr. 

4. Solve the equation cos 3^ + cos 2^ + cos ^1 = 0. 

5. Find all the values of 6 which satisfy 
(i) cos(9 + cos2^ + cos3^ + cos4^ = 0; 

(ii) sin3<9 + sin4^ + sin5<9 = 0. 

6. Solve (i) sin {A + 30°j = 1 -^ ^2 ; 

(ii) V'-3 sin A + cos A = ^/2 ; 
(iii) sin J. + cos ^ = 1 ; 
(iv) sin A + ^/2^ cos ^ = 2 ; 

(v) ^/2 (cos OX + sin 3a^) = 1 ; 
(vi) a cos6 + b sin 6 = c (put a = r cos 0(, h = r sin 0(). 

7. Prove that (i) 2 sin-i ^ ^/2 = 90° ; 

(ii) 2tan-^| = tan-4. 

8. In any triangle show that 

i? (sin 2^ + sin 25 + sin 2 Cj = 2r (sin^ + sin J5 + sin C). 

9. In any triangle show that 

a^ COS. 2 B + b'^ cos2 A = a- + &'^-4rt& sin^sin 5. 

10. Prove the formula (& + c) tan 1{B-C) = {b-c) cot hA. 
Write down two corresponding formulae. 

11. Using the fact that 3 x 18° = 90°-2 x 18°, find the values of 
sin 18° and cos 18°. 

Give a geometrical method for determining sin 18°. 



172 SUM AND DIFFERENCE FORMULAE 

12. Simplify 

., /sin 4^ cos4vl\ , , , .^ ,, 
V sm J. cos^ / 
sin 5^ -sin 3^ 2 sin 5 (9 + sin 3^ 



(") .^.^^ , ...o^ + 



cos5^ + cos3^ sin2^ cos5^ — cos3^ 

13. D, E, F are the feet of the perpendiculars from A, B, C on 
the opposite sides ; P is the orthocentre. Prove that 

(i) ^P=2i?cos^; (ii) PZ) = 2PcosP cos C; 
(iii) perimeter of triangle DEF = 4P sin ^ sin B sin C. 

14. State the general formula for all angles having a given 
cosine. 

Solve sin 3 ^ + sin 5 ^ + sin 7 ^ = 0. 

15. Find sec {A + B) in temis of the secant and cosecant of .4 and 
5, and prove secl05° = 72 (1 + ^3). 

16. Prove that 

sinl8° = |(v^-l) ; and that sin^ 30° = sin 18° sin 54°. 
Show that in any circle the chord of an arc of 108° is equal to 
the sum of the chords of arcs of 36° and 60°. 

17. Given cos ^='28, determine the value of tan ^.4, and 
explain fully the reason of the ambiguity which presents itself in 
your result. 

18. Prove that 

cos~'ii; + cos~^y = sin"^ {x \/\ —if^-y^/X —x^), 
and solve the equation 

tan-i{(a:+l)-f(a^-l)}+tan-i{(a^-l)-^a;} = tan-'(-7). 

19. Express sin3^ -r (sin2^ — sin^) in terms of cos ^. 

20. Prove the identities : 

(i) (l + cos^)tan2i.4= 1-cos^; 
(ii) (sec ^ + 2 sin A) (cosec ^ - 2 cos ^) = 2 cos 2 A cot 2 .-1. 

21. In any triangle prove that [h — c) cos ^A = a sin I {B — C). 
Ii A = 80°, a = 10, h-c = 5, find B and" C. 

22. Prove the identity cos 2^; sin 3a7 = sin x cos ix + cos .r sin 2.r. 

23. Solve the equations cos 2^ = cos(^-rt) ; cos 3^ = sin (6-^). 

24. (i) If the equation of a straight line is put in the form 
ij = mx + c, what is the geometrical interpretation of m ? 

(ii) Show how to find the angle between two lines whose 
equations are y = mx + c, y = rnx + c\ 



SUM AND DIFFEKENCE FORMULAE 173 

(iii) Deduce that the lines are at right angles if mm' = - 1 ; 
and parallel if m — m'= 0. 

(iv) Prove that the lines whose equations are ax + hi/ + c = 0, 
ax + Vy + c^O, are perpendicular if aa +W =0y and parallel 
if a/ a' = h/h'. 

25. Find the angle between the lines whose equations are 
(i) Sx-4:y=b, 4x-2ij = 7; 

(ii) 4a; + 3?/ = 6, Sx-4i/ = 9', 

(iii) 2x— y =3, 4a7 + 5?/=l; 

(iv) 2x-y =3, 4x + 2?/ = 5; 

(v) 2a; + 4i/ = 5, a; + 2t/=3. 

In each case verify by drawing to scale. 

26. Find the equations of the straight lines drawn through the 
point (3, 5), and respectively parallel and perpendicular to the 
line whose equation is 3ic — 4f/ = 5. 

27. Find the equation of the straight line, parallel to the line 
whose equation is iccos a+i/sin a =_p, and passing through the 
point [x, y). Deduce that the length of the perpendicular from 
[x', y') to the line x cos 0( + y sin oc = p i& x' cos Oi + ?/'sin Oi —p. 

28. Find in its simplest form the equation of the line joining 
the points {acos(CX + ^), &sin(CX + ^)}, {a cos (a -/3), &sin(a-/3)}. 

29. Prove that sin 55° sin 15° - sin 50° sin 10° - sin 65° sin 5° = 0. 

30. Show that in any triangle 

a^sin(J9-C ) &^sin(C-^) c^sin(J . -B) 
h + c c + a a + b 

31. If 2cos^ = ic + l/a; and 2 cos = 2/ + 1/^/, prove that 
2 cos {d + (f)) = xy + 1/xy and 2 cos {d — (p) = x/y + y/x. 

82. If d + (f) = 240°, and versing = 4 versing, find the values of 
6 and 0. 

33. Draw a curve to represent the variations in sign and magni- 
tude of (sin^->v/3cos^)-7-(-/3sin^ + cos^), from <9 = to e = n. 

34. If oc and /3 are the roots of « sin ^ + & cos ^ + c = 0, prove that 

cos-|((X4-3) _ COS |(CX-^) _ sin^(0( + /3) 
b —c a 

35. Eliminate 6 and from 

a sin ^ + & sin = h, (i) 

rt cos ^ — & cos (p = k, (ii) 

, cos {d + (f)) = I. (iii) 



174 SUM AND DIFFERENCE FOEMULAE 

36. Eliminate 6 and when two equations are the same as 
(i) and (ii) in Ex. 35, and the third equation is (i) sin(^ + 0) = I, 
(ii) tan(^ + 0) = /. 

37. Eliminate B and from the equations 

a cos{(b-hOC) sine/) . ,. x , • /, 

h cos(^-a) sm ^ ^ ■ v-r / 

38. Expand sin 5^ in terms of sin^, and cos 6^ in terms of 
cos^. 

39. If sinJ5 is the arithmetic mean between sin^ and cosyl, 
prove that cos2^ = cos2(^ + 45°). 

40. If a cos ^ + & sin B = c, show that 



B = tan-^ h/a + cos-i cj{ Va" + ¥). 

41. Find the maximum and minimum values of 

acos^ + Z>sin^ = c. 
Verify your answer when a = 3, Z> = 5, by drawing a graph. 

42. Prove that 

(i) sin B + sin 2B + sin 3 ^ + ... to n terms 

sin|«^sin|«4- 1 B ^ 
^ sinT^ " ' 

(ii) cos ^ + cos (^ + 5) + cos (^ + 2 i?) + . . . to n terms 

s,va.\nB C0& {A + \n~ 1 B) 
~ sin \B 

(iii) cos a + cos (a /3) + cos(a + 23)+ ...to nterms = 0, if 
n^ = 2n. 

43. Find the sum of n terms in the following series : 
(i) sin^ A + sin2 {A + B) + sin^ {A-v2B)+ ...; 

(ii) cosM + cos2(^ + jB) + cos2(^ + 2^)+...; 
(iii) sin^ sin 2^ + sin 2^ sin 3^4 + sin 3 J. sin 4-1 + .... 



REVISION EXAMPLES 175 



Revision Examples C. 

(All the following examples are taken from recent Examination 
Papers.) 

1. Find, without reference to the tables, the values of (i) sin 45° ; 
(ii) cos 150^; (iii) the tangent of the obtuse angle whose sine is 
1/v/lO. 

2. Trace the graph of the function cos^ + 2sin^ between the 
values and 180° of 6, and determine from your figure the value 
of 6 for which the function (ij is greatest, (ii) is decreasing most 
rapidly. 

3. Express tan 6 in terms of sec 6. 

Show that (sin 6 - cos d) (sec 6 + cosec 6) = tan 6 - cot 0. 

4. Prove that the sines of the angles of a triangle are in the 
ratios of the sides opposite them. 

5. Solve the equation 2 cos a; + sin a? = 2. 

6. In a right-angled triangle ACB, C being the right angle, 
the angle A is 35°, the side AB is 10 inches ; find the other sides. 

7. If cos(^ + -Bj = cos^cos5-sin.4sin5, calculate cos(A + B) 
when A = 50° and B = 50°. 

8. If OC is measured in radians, 

sin a = a-aV,3 + aV|5_-a7l2+ ... 
where j 5 means 1x2x^3x4x5. Find sin a correct to four 
significant figures when oc = 0'3. What is the angle OC in degrees ? 



9. Define the tangent of an angle in such a way that your 
definition is true for all angles. 

If 6 be an acute angle, prove that cos (90 + ^) = -sin 0. 

10. Arrange in. order of magnitude the angles 

2 sin~^ -51, i cos-i '32, tan'^ 8-9. 

11. Draw the graph of cos a; for values of x lying between 
0° and 90°. 

Use your figure to solve roughly the equation .r = 100cos:r°, 
and verify your solution by the tables. 



176 REVISION EXAMPLES 

12. Given that sin 20° = O'Si and cos 20° = 0*94, write down 
the values of sin 160° + cos 160°, of sin 250° + cos 250°, and of 
sin 340° + cos 340°. 

13. In any triangle ABC, show that 
(i) c = acos^ + &cos^ ; 

(ii) c2 = a2 + &2-2a&cosC. 
Find c when a = 5, & = 6, and 0=155° 31', having given 
cos24°29' = 0*91. Verify your result by a diagram drawn to 
scale. 

14. Find to the nearest degree the angle subtended at a man's 
eye by a tower 50 feet high, when the man has stepped back 
30 feet from the tower, assuming the height of his eye above the 
ground to be 5 feet 6 inches. 

15. Write down a formula for sin^J. in terms of the sides of 
the triangle ABC and explain the notation. How is the formula 
modified when h — c^ 

Given that the sides are 100, 200, 160 units in length, calculate 
the smallest angle. 

16. A and B are two acute angles but A + B \^ obtuse; prove 
that cos {A + B) = cos A cos i? — sin ^ sin B. 

Solve completely cos x + ^mx = cos Oi — sin OL. 



17. Define the tangent of an angle, and show geometrically that 

tan^tan(90° + ^) + l = 0. 

18. Draw a circle of diameter 1 inch. Draw a diameter AB 
and the tangent to the circle at B, divide either of the semi- 
circumferences between A and B into 8 equal parts, join A to 
the points of section, and produce the joining lines to meet the 
tangent at B. Measure the distances of the points so found from 
B, and use the results obtained for drawing the graph of tan ^4 
from ^ = 0° to ^ = 90^. 

19. Prove that the area of the triangle ABC is 

\a^ sin B sin C/sin A. 
Use this expression to find the area of the triangle when 
a = 106-5 yards, A = 56° 37', B = 75° 46.' 



REVISION EXAMPLES 177 

20. A person walking along a straight level road running clue 
East and West observes that two objects P and Q are in a line 
bearing North-West, and after walking a further distance d he 
observes that P bears due North and that the direction of Q 
makes an angle A with the direction in which he is walking. 
Prove that the distance PQ is cZcos^/sin (^-45°). Find PQ 
when d = 1372 yards, and the angle A = 56° 31'. 

21. (i) Show that (sin^ + cos^)2 + (sin^-cos^)2 = 2. 

(ii) Considering only values of A between 0° and 90°, find the 
value of A when sin^ cos J. has its greatest value, and show that 
the same value of A gives the greatest value of sin ^ + cos ^. 

22. Let AD bisect the angle ^ of a triangle ABC, and let it 
meet BCin D -, show that BD sin B = CD sin C. 

Hence show that BD.AC= DC. AB. 

23. (i) Show geometrically that 

sin {A + B) = sin ^ cos ^ + cos A sin B, 
when A, B, and A + B are each less than 90°. 

(ii) By means of this formula, and in view of the restrictions 
under which it has been obtained, show that 

sin 464° = sin 153° cos 311° + cos 153° sin 311°. 

24. Find tan 6 and x in terms of a and b from the equations 

a sin ^ + & cos 6 = ^x, 
« cos ^ — 2 & sin ^ = 2 37. 



25. An angle is made to increase gradually from 0° to 360° ; 
state briefly how the values of its sine and of its cosine change 
during the increase of the angle. 

26. Calculate the values of A between 0° and 360° for which 
tan J. -2 cot ^ = 1. 

27. A and B are two milestones on a straight road running due 
East across a horizontal plane, C an object on the plane. The 
bearings of C as viewed from A and B are 35° North of East, 
and 55° North of West respectively. Find, to the nearest foot, 
(1) the distance of C from A, (2) the distance of C from the 
nearest point of the road. 

1916 M 



178 REVISION EXAMPLES 

28. Plot in relation to the same axis and origin the values of 
tana; and 2 sin a; for the values 0°, 12=^30', 37° 30', 50°, 62° 30', 
75° of X, draw the graphs of tan x and 2 sin x, and find from them 
the values of x for which tana? = 2 sin a;. Give the general solu- 
tion of the equation tan a? = 2 sin x. 

29. Prove that (cos A + sin A) -h- (cos A - sin A) = tan {A + 45°). 

30. Prove for a triangle in which the angle B is obtuse the 
relation sin Bjh = sin C/c, and deduce the relation 

tan ^{B-C} = {b- c)/{b + c) cot I A. 
If & = 27*3 yards, c = IS'S yards, A = 48° 36'", find B and C. 

31. Prove that in a triangle ?• = 4 J? sin ^ J. sin 1 5 sin | C 
ABC is a triangle ; B'C is drawn through A parallel to BC, A'C 
through C perpendicular to AC, and A'B' through B perpendicular 
to AB. Prove that the area of the triangle A'B'C is 

\ or cos- {B — C)-ir cos B cos C sin A. 

32. (i) Find sin ^ + sin 5 in terms of functions of half the sum 
and of half the difference of the angles A and B. 

(ii) If ^ + 5 is between 90° and 180°, find under what circum- 
stances tan J. + tan 5 will be negative. 



33. Find to the nearest minute the angle of a regular polygon of 
17 sides. 

What angles less than 360° satisfy the equation 
2cos2^-f llsin^-7 = 0? 

34. Prove the identity 

tan^^ coi^A _ 1 — 2 sin-.l cos-.l 

1 + tan^^ 1 + cot^^ sin J. cos ^ 

35. Assuming the formula a^ = h'^ + c^ — 2 he co^ A, establish a 
formula for tan^^ in terms of the sides of the triangle, and find 
the greatest angle of the triangle whose sides are 13, 14, 15. 

36. Prove that for any triangle ABC 

rt/sin A = &/sin B = r/sin C. 
U B = 39° 17', a = 4*2, and b = 3-5, solve the triangle fully ; 
draw a figure to illustrate your solution. 

37. The angles of elevation of a vertical pole from two points on 
a horizontal line passing through its liase and 6 feet apart are 
a and /3 ; prove that the height of the pole is b/[cot 0( — cot^) feet. 



KEVISION EXAMPLES 179 

38. From a point on a horizontal plane passing through the foot 
of a tower the angles of elevation of the top and bottom of 
a flagstaff 20 feet high, placed vertically at the summit of the 
tovrer, are 51*2^ and 47"3''. Find the height of the tower. 

39. Prove that (i) sin (A + B) = sin AcosB + cos A sin B ; 

(ii) cos2^{l + tan2^)= l-tan-.4. 
Use (ii) to find the value of tan 15°. 

40. Reduce the fraction a -r (cos^^ — sin-^; to a form suitable 
for logarithmic calculation, and perform the calculation when 
« = 10, A = 29' 55', and B = 15" 5'. 



41. Prove that sin*^ .1 + cos'^ .^ = 1 for all values of A less 
than 180°. 

A and B are each less than 180^ sin ^ = '3900, smB= -9208, 
find four possible values of A + B. 

42. Find from your tables the value to two decimal places of 
the expression sin^ + sin2^, when B is 10^ 20°, 30°, ... 90', and 
from these draw a graph of the expression on a suitable scale. 

43. In a triangle ABC prove that 
(i) 2bc cos A = b- + c^- a-; 

(ii) cosM + cos2-B + cos^co35 = f, if C=60°. 

44. In a triangle a = 12*76, h = 10-87, c = 8-37, find C. 

45. Show how to find the distance between two visible but 
inaccessible objects. 

46. In any triangle ABC show that four times the area equals 
{(("^ + b- + c^) -^ (cotan A + cotan B + cotan C). 

Show also that when C is a right angle this expression reduces 
to 2a&. 

47. Prove the identities : 

(ij 1/sin 2.4 = 1/tan A - 1/tan 2.4 = tan A + 1/tan 2^ ; 

,.., . ^ , sin ^^ + sin 4 J. 

(ii) tan ?>A = ^^-p- T-, • 

^ ' co8 2J. + cos4^ 

48. What is the meaning of tan-* x ? 

Prove that tan"' a? + tan"'?/ = id,n-'^[{x + y) -f {\-xy)]. 
Prove that 45° is one value of tan~^ -| + tan"' |- + tan"* j^jj. 



49. Prove that (i) secM = 1 4- tan^ ^ ; 

(ii) cosec^-cot^ = tan^^. 

M 2 



180 REVISION EXAMPLES 

50. Construct an angle whose sine is 0'76. From your figure 
obtain the value of the cosine of the angle. 

51. On squared paper draw graphs of tan^ and cot ^ between 
^ = 10° and 6 = 80". From the graph, or otherwise, find angles 
which satisfy the equation tan 6 + cot ^ = 3. 

52. Let D be the point in which one of the escribed circles 
touches the side BC of a triangle ABC. If the sides a, h, c of 
the triangle are given, find expressions for the radius of that 
circle and for BD and CD. 

53. A tree which grows at a point A on the north bank of 
a river is observed from the points B and C on the south bank. 
The distance BC is 200 metres, the angle ABC is 46° 80', and the 
angle ACB is 58° 20'. Calculate the distance of A from the 
straight line BC. 



54. Prove the formula sin|J. = \/{s — h) (s — c) -r he. 
If, in a triangle ABC, 2b = a + c, prove that 

sin|5 = 2 sin 1^ sin I C 

55. Find the angles B and C and the radius of the circum- 
scribed circle of a triangle ABC in which A = 32° 42', a = 36, 
6 = 44. 

56. State De Moivre's Theorem, and, assuming it for integral 
indices, prove it for fractional indices. 

Write down all the values of (>/— l)o. 



57. U A is an obtuse angle whose sine is i%, find the values of 
cos^ and tan^. 

58. (i) Show, by drawing graphs of the two expressions sinu; 
and cos (.r + 90°), that sina;= —cos (a; + 90°). 

(ii) If sin a? = ^ \/2, find a formula which gives all the values 
of .r which satisfy the equation. 

59. Prove that in a triangle 

(i) tan ^B = ^{s -c}{s-a}-^s {s — b) ; 
(ii) b cos B + c cos C = a cos {B — C). 



EEVISION EXAMPLES 181 

60. If two sides of a triangle and the angle opposite one of them 
are given, show how to solve the triangle, and discuss by the aid of 
a figure all the cases that can arise. 

One side of a triangle is 20 inches long, the opposite angle is 
34° 42' ; another side is 30"41 inches. Find the sides and angles of 
the two possible triangles. 

61. Assuming the formulae for the sine and cosine of half an 
angle of a triangle in terms of the sides, prove that 

(i) r = ^/{s — a){s — b) ( s — c) -r s ; 
(ii) Z?= a/2 sin ^. 

62. I observe the altitude of an airship to be 35°, and that of the 
sun, which is in the same vertical plane as my eye and the airship, 
to be 40^ The shadow of the airship falls on a tree on the same 
level as my eye and 500 feet in front of me. Find the height of 
the airship. 

63. In any triangle prove that 

sin ^ - sin ^ + sin C = 4sin ^vi cosi^sin^C 
Assuming the formula for expanding ti\.n{A + B), find expressions 
for tan 2^ and tan 3 J. in terms of tan J.. 

64. Make an angle AOC and bisect it by the line OB. From 
any point A in OA draw ABC perpendicular to OB, meeting 
OB, OC in the points B and C respectively, and draw AX perpen- 
dicular to OC. Use this figure to prove that 

(i) sin2J.<2sin^; (ii) tan 2 xt > 2 tan .4. 



65. Prove that sin^^ + cosM = 1. 

Having given that the sine of an angle is '56, calculate its cosine. 

66. Show how to construct an angle whose sine is "6. 
Find a value of x which satisfies the equation 

4 sin ic + 3 cos a; = 1. 

67. Given two sides of a triangle and the included angle, show 
how to find the remaining side and the other angles. Prove such 
formulae as you require. 

If a = 1097 feet, b = 781 feet, C = 31° 30', find c to the nearest 
foot. 

68. A ship is sailing at the rate of 7 miles an hour. A man 
walks forward across the deck at the rate of 4 miles an hour 



182 KEVISION EXAMPLES 

relative to the deck, in a direction inclined to the keel at an angle 
of 60°. Find the direction of his actual motion in space. 

69. Prove the formula cos {A-B) = cos AcosB + sin A sin B. 
Show that if xy = a'^ + 1 then 

cot~^ {a + x) + cot"' (« + y) = cot"' a. 

70. Find an expression for cos{Oi. + ^ + y) in terms of sines and 
cosines of Of, 0, and y. 

Prove the identity 

cos OC cos + 7) + COS /3 cos {y + OC) + COS y COS {01. + ^) 

= COS (ex + /3 + -y) -h 2 COS OC COS /3 cos y. 

71. At what angle must forces of 4 dynes and 5 dynes act so 
that their resultant may be a force of 6 dynes ? 

72. If B be the circular measure of an angle, prove that, as 6 is 
indefinitely diminished, the ratios 6 : sin d, 6 : tan 6 approach to 
the limit unity. 

A man standing beside one milestone on a straight road observes 
that the foot of the next milestone is on a level with his eyes, and 
that its height subtends an angle of 2' 55". Find the approximate 
height of that milestone. 



73. Write down the values of sin 36° and cos 36° as given by 
your tables. Calculate the sum of the squares of these numbers 
to six decimal places, and explain why the result differs from unity. 

74. Give definitions of the tangent and cotangent of an angle 
which is greater than 90° and less than 180°. 

Prove that (i) tan (180 - ^) = - tan ^ ; 
(ii) tan(90 + <9)=-cot^. 

75. In any triangle prove that a/sin A = hj&m B = c/sin C. 

If BC be 25 inches, and CA be 30 inches, and if the angle ABC 
be twice the angle CAB, find the angles of the triangle ABC, and 
show that the length of the third side is 11 inches. 

76. F, Q, R are three villages. P lies 7 miles to the North-East 
of Q, and Q lies 11^ miles to the North-West of B. Find the 
distance and bearing of 7' from B. 

11. A point is moving with velocity 50 feet per second in 
a direction 60° North of East. Find the resolved parts of the 
velocity in directions East and North. 



EEVISION EXAMPLES 183 

78. A man has before him on a level plane a conical hill of 
vertical angle 90^. Stationing himself at some distance from its 
foot he observes the angle of elevation Oc of an object which he 
knows to be half-way up to the summit. Show that the part 
of the hill above the object subtends at his eye an angle 

, tan0c(l-tan3() 

tan~ — • 

1 + tan DC (1 + 2 tan y) 

79. The latitude of London is 51' N., and the radius of the 
Earth 4000 miles. How far is London from the Equator measured 
along the Earth's surface, and how far from the Earth's axis ? 

80. Prove that sin A + sinB = 2 sin l (A + B) cos I {A - B). 
Show that sin 10' + sin 20' + sin 40' + sin 50' = sin 70' + sin 80°. 



MISCELLANEOUS PKOBLEMS 

(The following examples are taken from recent Army Entrance 
and Civil Service Papers.) 

1. I take measurements to determine the air space of a rect- 
angular hall : length 18*4 metres, breadth 11*8 metres, inclination 
to floor of diagonal of side wall 31 "8°, of diagonal of end wall 44°. 
Calculate the air space. 

More measurements were taken than were necessary. Check 
the measurements by deducing one of them from the other three. 

2. The ancient Greeks measured the latitude of a place by 
setting up a vertical rod and comparing its length with the 
length of its shadow. Supposing observations taken at mid-day 
at the equinox (when the sun is vertical at the equator) to give 
^ as the ratio of the rod to shadow at Alexandria, and ^ as the 
ratio at Carthage, find the latitude of each place. 

3. The following method of determining the horizontal distance 
PR, and the difference of level QR between two points P and Q, 
is often used. A rod with fixed marks A, B on it is held vertical 
at Q, and the elevations of these points, viz. ACD = (X, BCD = jS, 
are read by a telescope and divided circle at C, the axis of the 
telescope being a distance CP=a above the ground at F. If 
QA = h, and AB = s, write down expressions for PB and QB. 
Find PB and QB when 0( = 6° 10', 8 = 1° 36', the values of a, h, 
and s being 5 feet, 2} feet, and 5 feet respectively. 

4. Three balls, 5 cm. in diameter, lie on a floor in contact, and 
a fourth equal ball is placed on them. Find the height of the 
centre of the fourth ball above the i)lane of the other three 
centres. Find also the inclination to the vertical of any line 
that touches both the top ball and one of the lower balls. 

5. The curved surface of a right circular cone whose semi- 
vertical angle is 45° is made by cutting out a sector from a circular 
sheet of copper, the diameter of the sheet being 5C cm. Deter- 
mine the angle of the required sector. 



MISCELLANEOUS PROBLEMS 185 

6. If tangents be drawn to the inscribed circle of a tiiangle 
parallel to tbe sides of the triangle, show that the areas of the 
triangles cut off by these tangents are inversely proportional 
to the areas of the corresponding escribed circles. 

7. A rod BC, of length 5*8 cm., rotates about B. Another rod CA, 
of length 8*6 cm,, has one end C hinged to the first rod, while the 
other end A slides along the line BO. By drawing the rods in 
various positions, find how the length o^ BA varies as the angle B 
increases ; and show BA as a function of angle 5 in a graph for 
one revolution of BC, showing the actual length of BA and repre- 
senting 30° by 1 cm. 

Write down an equation connecting the angle B and the lengths 
of the three sides of the triangle ABC. Solve the equation to find 
the length of BA when angle B is 35"". 

8. The extreme range of the guns of a fort is 8000 metres. 
A ship, 14000 metres distant, sailing due East at 24 kilometres an 
hour, notices the bearings of the fort to be 20° 30' North of East. 
Find, to the nearest minute, when the ship will first come within 
range of the guns. 

9. The face of a building is 136 feet long. A photographer 
wants to take the building from a point at which the face subtends 
an angle of 37°, and for this purpose he starts off from one corner 
of the building in a direction making an angle of 127° with the 
face in question. Find by calculation the distance from the corner 
at which he must take the photograph. Calculate the area of 
ground in the triangular space between his position and the face of 
the building. 

10. From the top of a telephone pole three wires radiate in 
a horizontal plane. One wire, A, exerts a tension of 100 lb. 
weight ; the next, B, makes an angle of 90° with A and exerts 
a tension of 80 lb. weight ; the third, C, makes an angle of 35° 
with B and an angle of 125° with A, and exerts a tension of 
90 lb. weight. It is required to equilibrate the three tensions by 
means of a fourth wire. Find its direction and tension. 

11. A man passing along a straight road measures the angle 
between the direction of his advance and a line drawn to a house 
on his left. At a certain moment the angle is 36° 21'. He walks 
on 1500 yards and finds that the angle between the same direction 



186 MISCELLANEOUS PROBLEMS 

and the line to the house is now 125° 36'. Find the distance of 
the house from the road. 

12. Plot a curve giving the sum of 4sin^ and 3 sin 2^ from 
S = 0° to 6 = 180°, and read off the angles at which the greatest 
and the least values respectively of this sum occur. For the angle 
use 1 cm. to represent 10 degrees, and for 4 sin ^ + 3 sin 2^ use 
1 cm. to represent unity. Also estimate the slope of the curve 
when 6 = 90° and when d = 135°. 

13. A, B, and C are three buoys marking the corners of a 
triangular yacht racecourse round an island. The angles A, B, 
and C of the triangle ABC are found to be 75°, 63°, and 42° 
respectively. P is a flagstaff on the island, from which A and B 
can be seen, and the distances of P from A and B are found by 
a range-finder to be 650 yards and 585 yards resi3ectively, and the 
angle APB to be 187°. Calculate the length of one lap of the 
course. 

14. Draw an angle XOP of 30°, making OP 2" long : through P 
draw PQ parallel to OX and in the same direction : produce XO to 
X\ making OX' = OP, and join X'P: cut off PQ = PX. Join OQ 
and measure the angle XOQ carefully. Now denote XOQ by 0, 
XOP by 6, and OP by e, and write down an expression for the 
length PQ. Deduce an equation for 6 and 0, and solve it for tan (p. 

Use your tables to evaluate (p when 6 = 30°, and compare your 
result with the measured value. It is said that the given con- 
struction trisects an angle. What is the percentage error for 30° ? 

15. In running a survey the lengths of a series of lines are 
measured, and the angle each line makes with the direction of 
magnetic Noiih is measured by a theodolite. The data booked are 
given in the table below : — 

Line. Length in feet. Bearing. 

AB 433 29° 15' 

BC 521 89° 12' 

CD 352 182° 38' 

DE 417 233° 25' 

The angles are measured clockwise from the magnetic North 

direction. 

By an error the measured length of the closing line LA of the 

survey was not recorded, nor its bearing; from the data given in 

the table calculate these missing data. 



MISCELLANEOUS PROBLEMS 187 

16. AOB and COD are two straight roads crossing one another 
at an angle of 57". A motor-car, travelling at the rate of 18 miles 
an hour along AOB, is 1500 yards from 0, when a man, walking at 
the rate of three miles an hour along COD, is a quarter of a mile 
from 0; car and man are both approaching 0. Find graphically 
the motion of the car relative to the man. Hence find the least 
distance between the car and the man, and when they are at this 
distance from one another. 

17. In a triangle a = 10 cm., b = l cm., one angle is 95°. 
There being no restriction as to which angle of the triangle is 95", 
discuss how many distinct triangles can be made. Select any one 
case, and for this case calculate the remaining sides and angles. 

18. X and Y are two fixed points in a straight line, P a point 
which so moves that cosPA^Z+cos PYX = Jc (a constant). Prove 
the accuracy of the following construction for obtaining the locus 
of P: With X and Y as centres describe circles of radius XY/k. 
From any point X in XY draw XAB perpendicular to XF cutting 
the former circle in A and the latter in B. Draw XA and YB, 
intersecting in P. Then P is a point on the locus. 

19. A candle, C, is placed on the floor at a distance r from 
a point on a wall, and at the same level as the candle-flame, 
and the angle which OC makes with a perpendicular to the wall 
at 0, is 6. The illumination received on the wall at from the 
candle is known to be equal to Acosd/f- where ^ is a constant. 
If the candle be moved about on the floor in such a way that this 
illumination remains constant, plot on a diagram the curve 
described by the candle-flame. 

20. Two small islands are 5 miles apart, and there is known to 
be a rock distant 3 miles from each. A ship is in such a position 
that the islands subtend an angle of 66^ at the ship. Calculate, 
to the nearest hundredth of a mile, her least possible distance 
from the rock. 

21. Find by means of a graph two acute angles 6 for which 
5sin2^ = 3sin(9 + 2-5. 

Find also the greatest value of 5 sin 2^ -3 sin ^ when 6 is an 
acute angle, and the angle to which this value corresponds. 

22. The elevation of an aeroplane which is flying horizontally 
on a flxed course at a height of 150 feet is taken at two instants 



188 MISCELLANEOUS PROBLEMS 

at an interval of 20 sees. At the first observation the elevation 
is 10° and the bearing is clue North, and at the second the 
elevation is 6|° and the bearing is N. 35° E. Find the course and 
speed of the aeroplane. 

23. The strength of an electric current C is obtained from the 
formula C = k tan 6 where 6 is the angle read off in degrees on 
an instrument, and k is n constant. If an observer makes an error 
of 8^ in reading the angle ^, prove that the value of C thus 
obtained will be wrong by an amount equal to J^tt Ccosec2^S^. 
Hence find the error per cent, in C produced by making a mistake 
of -j^j degree when 6 is 60°. 

What value of B is likely to produce the smallest error in the 
value of C ? 

24. If P denote the pressure of wind in lb. per square foot on 
a plane surface at right angles to the direction of the wind, and 
2) denote the normal pressure of wind in lb. per square foot on 
a plane surface inclined at an angle 6 to the direction of the 
wind, the following formulae are used to determine the ratio p : P. 

(i)iV-P=(sin^)^'^^'-''''^-^; 
(ii) p/P = 2 sin 6/(1 + sin^ S). 
Compare the values of p/P given by these formulae for the 
values 10° and 50° of d. 

25. A man walks due W. from a point ^ up a straight path 
inclined at 10° to the horizon. After walking 2 miles he reaches 
B, and turns up another straight path to the NE., sloping 15° 
upwards. He reaches C after walking one mile from B. What is 
the distance in a straight line from C to ^ ? What is the height 
of C above the level of A ? Taking the face of the hill ABC 
as a plane surface, what is the greatest slope ? 

26. A flagstaff stands vertically on horizontal ground. Four 
ropes, each 56 feet long, are stretched from a point in the flagstaff', 
50 feet above the ground, to four pegs in the ground, arranged 
at the corners of a square. Calculate the angle between two 
adjoining ropes. 

27. Q is the centre of a circle of radius 10 cm., and QO is 
a radius. The seven points ABC ... lie on the circumference and 
the angles OQA, OQB, OQC... have the values 10", 20", 30°... 70°. 
Find by drawing or calculation the lengths of the chords OA, 
OB, OC..., and tabulate the results. 



MISCELLANEOUS PROBLEMS 189 

Draw a graph to give the length of chord of the circle in terms 
of the angle which it subtends at the centre (for angles up to 70"). 
Show the chord's actual size, and represent 4 degrees by 1 cm. 

From jour graph find the length of the chord which subtends 
an angle of 48°. Make a triangle having one side of this length, 
and the other two sides 10 cm. long, and therefore having an 
angle of 48''. 

Check the accuracy of your drawing by measuring this angle. 

28. A square made of jointed rods each 4 inches long is 
deformed into a rhombus having half the area of the square- 
Calculate the lengths of the diagonals of the resulting figure and 
check by drawing. If it is part of a lattice-work, the original 
height of which is 6 times the diagonal of one of these squares, 
find by calculation how much the height of the lattice-work could 
be increased if each square were reduced to half its area. 

29. A straight rod AB, 3 feet 9 inches long, is held under water, 
A being 2 feet 6 inches and B 9 inches below the surface. Calcu- 
late (a) the distance below the surface of a point C on the stick 
which is 12 inches from A, (h) the angle which the stick makes 
with the surface of the water. 

If a parallelogram is held under water, show that in every 
position the sum of the depths of the 4 corners is 4 times the 
depth of the point of intersection of the diagonals. 

30. If a closed loop of thread is placed on a soap-film that 
covers a ring of wire, and the film within the loop is joierced, the 
film outside takes up as small an area as possible and thus pulls 
the thread at A into a circle. Calculate the diameter and the 
area of the circle formed by the thread if length of thread forming 
the loop is 6 cm. 

If the ends BC of the thread are attached to the ring, and the 
film on one side of the thread is pierced, the thread again becomes 
a circular arc. If the thread BC is 6 cm. long, and the angle 
it subtends at the centre of the circle of which it forms an arc 
is 120°, calculate the length of the chord BC, 



EXAMINATION PAPERS 

OXFOKD AND CAMBRIDGE SCHOOLS' EXAMINA- 
TION BOARD. 

School Certificate, 1910. 

1. Define the tangent of an angle. 

Construct an acute angle whose sine is "6, and find its cosine and 
cotangent. 

2. Prove that cos (180 -a) = -cos Of. 

Arrange the angles Oi, /3, y in order of magnitude, if 

sin 3^ = -8211, cosi3 = -7738, tany= -O'GlOi, 
the angles being positive and each less than 180°. 

3. What is the length of the shadow of a man, 5 feet 8 inches 
high, cast by the sun when its altitude is 55° 30' ? 

4. Draw the graph of 10 + 10 cos 2a; for values of x between 
0° and 60°. Find a value of x to satisfy the equation 

a; = 10 + 10 cos 2 0^°. 
[Take one-tenth of an inch as unit along both axes.] 

5. Prove that in any triangle sin A/a = sin B/b. 

U A = 63°, B = 49°, a = 50 inches, find b to the nearest tenth 
of an inch. 

6. Prove that 

... cos ^ + sin ^ cos ^- sin ^_ 2 

^^ cos ^ -sin ^ cos(^ + sin^ ~ l-2sin-^' 
(ii) (sec 6 + tan 3) (cosec 6 ~ cot 6) = (cosec ^ + 1) (sec ^ - 1). 

7. If 2 sin ^ + 5 cos ^ = 5, prove that tan^ = or 20/21. 

8. Prove that sin {A — B) = sin A cos B — cos A sin B, where A 

and B are both acute angles and A is greater than B. 

^ , , , sin 5 ^ + sin ^ , _ » „ ^ 

Prove that . ^ , ; — - = 1 + 2 cos 2 -4. . 

sind^ —sin J. 

9. Show that in any triangle ABC 

b + c _ coslJB-C) 
a sin h A 

If fc + c = 24'8 cm., a = ir89 cm.,''yl = 39°, find B and C. 



EXAMINATION PAPERS 191 

10. A lighthouse is observed from a ship which is steaming due 
N. to bear 62= W. of N. ; after the ship has sailed 10 miles the 
lighthouse is observed to bear 40° W. of S. Calculate the distance 
of the ship from the lighthouse when it was nearest to it. 

Higher Certificate, 1910. 
Part I. 

1. Give a definition of cos^ that holds for all angles from 0' to 
180°. Show that cos ( 180 -d)= - cos 0. 

2. Show that secM = 1 + tan2^. 

Draw the graph of 1 +sin dx°, where x lies between 0° and 60°. 

3. Construct an acute angle whose cotangent is 2, an obtuse 
angle whose sine is 3, and an obtuse angle whose secant is —3*5. 
Measure these angles as accurately as you can with the protractor, 
and verify your results by means of tables. 

4. (i) Verify that 30'', 45°, and 60'' are solutions of the equation 

sin3a; + cos3rr = 2 cos 2a;. 
(ii) Show that 
(cosec^ + sec^j^ + (cosec A - sec A)^ = 2 cosec'^ (3 sec^A — 2). 

5. Show that in an obtuse-angled triangle 

sin A/a = sin B/h = sin C/c. 
A man observes that the angular elevation of the foot of a 
tower on a distant hillside is OC, and that the angular elevation 
of the top of the tower is /3, and he knows that the height of 
the tower is h feet. Show that his horizontal distance from the 
tower is ^ cos (X cos /3 cosec (/3 — 3ii). 

Part II. 

6. Draw the graph of cot.r between the values —180 and + 180 
of X, taking the unit of x to be -^q inch and the unit of y to 
be one inch. 

Find an acute angle to satisfy the equation x = 60 cot a;''. 

7. Show that sin (^-5) = sin .4 cos ^- sin 5 cos. 1, taking A 
and B to be acute angles of which A is the greater. 

If tana; = A; tan (^ -a;), show that 

{k-l)smA= {k-hl)3in{2x-A). 
Use this result and tables to solve the equation 
tana: = 2 tan (50° -a;). 



192 EXAMINATION PAPERS 

8. In the triangle in which a = 72 feet, B = 40°, and C = 55°, 
find c. 

9. Find in terms of a, b, and c the radius of the circle escribed to 
the side BC of the triangle ABC. 

If Jj is the centre of this circle, show that 

aAI,''-hBI,^-cCI,' = abc. 

10. AB is a diameter of a circle whose centre is ; on AB an 
equilateral triangle ABC is described, and a point D is taken in AB 
such that 1BD = 2AB ; CD is produced beyond D to meet the circle 
at E. Show that tan ADC = 7/^/3 and that sin OED = 3/^52. 

Hence, or otherwise, show that the error made in taking the arc 
BE to be one-seventh of the circumference of the circle is less than 
'2 per cent. 

Part III was beyond the scope of this book. 



OXFORD LOCAL EXAMINATIONS. 

JUNIOE. 1910. 

1. (i) Find the sine of 60° ; 

(ii) If -4 is an acute angle, and cos^ = ^, find the value of 
4 tan ^ + 5 sin A. 

2. P and Q are points on a straight stretch of a river bank and R 
is a point on the other bank. If cot PQR = '32, cot ^Pi^ = '43, 
and the length of PQ is 15 yards, find the breadth of the river. 

3. Draw the graph of sin (45° + 2a:) between x = and 180^ 

4. U A, B, A — B are all positive acute angles, prove that 

cos {A — B) — cos AcosB + sin A sin B. 

5. (i) A, B, C are the angles of a triangle; if tan^ = ^ and 
tan ^ = ^, find the angle C. 

,.., „ ,- , cos5u4 + cos3^ , . 

(u) Prove that - — p— : — ir—r = cot A. 

^ ^ sin 5 ^ - sm 3 ^ 

6. Solve the equation cos2^ + sin^ =?= 1, 



EXAMINxVTlON PAPERS 193 

7. Prove for any triangle that 

(i) rt/sin A = b/tiin B = c/sin C ; 
( ii) {b + c) cos A + {c + a) cos ^ + (« + 6j cos C = a + 6 -I- c. 

8. Find the angles A and i? of a triangle ABC in which « = 13, 
/> = 14, c = 15, having given: 

Iog2 = -o010, log? = -8151, 
Z tan 26= 34' = 9-6990, 
Z tan 29' 44' = 9-7569. 



Senior. 1910. 

1. Find the tangent of 30"". 

Using the values of tan 30° and tan 45'', prove that 
tan 75° = 2 + -\/3. 

2. A man on a straight level road observed two objects Pand Q 
{P being the nearer) in a horizontal straight line inclined to the 
direction of the road at an angle 0(. If tan Oi = -75, FQ = 400 yards, 
and the shortest distance of F from the road is 180 yards, what is 
the shortest distance of Q from the road ? 

3. Prove that cos3^ = 4cos^.4-3cos.l. Find sin 18°. 

4. If ABC is a triangle in which 6 = c = 5 inches and 
a = 8 inches, find the values of tan A and tan B. 

5. Prove that 

cos2 j^ ^ cos2 B = sin2 {A + B) + 2 cos A cos B cos {A + B). 

6. Prove that in any triangle c = (« + &) sin^, where 

cos ^ = 2 ^/ab cos | C/{a + &). 
In a triangle ABC, a = 36 feet, & = 4 feet, C = 55°. Using 
the above formula, find the third side, having given 

log 6= -7782, Z cos 57° 51' =9-7261, 
Zcos27°30' = 9-9479, sin 57° 51'= '8467. 

7. Find the radius of the circle inscribed in the triangle ABC. 
C is the centre of a circle of diameter d, and A, B are two 

points on the circumference of the circle. If I is the length of the 
chord AB and S is the diameter of the circle which touches 
CA, CB and also the arc AB at its middle point, prove that 
1/5 = 1/(^+1//. 

1216 N 



194 EXAMINATION PAPERS 

CAMBRIDGE LOCAL EXAMINATION. 
Junior. 1909. 

1. Define the sine of an angle. What are the greatest and least 
values which the sine of an angle can have ? 

Prove that sin^ = cos A x tan J., and that 

sin A sin B cot B = cos A cos B tan A. 

2. Construct an angle vrhose tangent is 1*45, and measure it 
with a protractor. Verify your results with the help of the tables. 

3. Prove that 

(i) sin A = tan A/{ -/l + tanMj ; (ii) cos (90° + A)= - sin A. 

4. Find by drawing graphs of sin^ and sin 2^ for what value 
of A, less than 90°, 2 sin ^ - sin 2 ^ = 1. 

5. A vertical post casts a shadow 15 feet long when the altitude 
of the sun is 50° ; calculate the length of the shadow when the 
altitude of the sun is 32°, 

6. Prove that sin ^ + sin i? = 2 sin 1{A + B) cos ^{A- B), and 
that tan 2 J = 2 tan A/{1 - tan-^). 

Show that sin ^ - 3 sin 3 ^ + 3 sin 5 ^ - sin 7 ^ = 8 sin^^ cos 4 A. 

7. Prove that, in any triangle ABC, acos,B+ h cos A = c. 
Show also that ftan A + tan B) (tan A — cot C) = sec"^. 

8. Show how to solve a triangle when three sides are given. 
Find the greatest angle of the triangle whose sides are 5'2 inches, 

77 inches, and 9*1 inches. 

Senior. 1909. 

1. Show that the ratio of the circumference to the diameter of 
a circle is an invariable quantity. 

Find to an inch the diameter of a wheel which makes 400 
revolutions in rolling along a track one mile long. 

2. Any positive proper fraction being given, show that there are 
two angles, one acute and the other obtuse, such that the sine of 
either is equal to this fraction. 

If the fraction is |, use the tables to find the angles, and the 
cosine and tangent of each. 

3. Find by aid of the tables the values of sin it- -tan 2x for the 
values 0°, 10°, 20°, 30°, 45°, 60° of x. 

Make a graph to give the values of sin it' — tan 2 jc from ^=0 to 
X = 60°. 



EXAMINATION PAPERS 195 

4. Show that sin A + sin 5 = 2 sin i (.1 + B) cos I (A - B). 
Prove also that 

(i) tan2^ = (l-cos2.4)^(l + cos2^); 
(ii) sin 55° sin 15° - sin 50° sin 10° - sin 65° sin 5° = 0. 

5. Find the greatest angle of a triangle whose sides are 15, 21, 
28 inches in length. 

Show that in any triangle 

a'^sin (^-C) &-sin(C-^ ) c=^sin(.l-^) 
h + c c-^a a + b 

6. Find an expression for the radius of the inscribed circle of 
a given triangle. 

Determine to one place of decimals the length of the radii of the 
inscribed circle, and of the escribed circle opposite the greatest 
angle of the triangle referred to in Question 5. 

Questions 7 and 8 were outside the scope of this book. 

(The two following questions may be taken instead of 7 and 8, but 
considerably lower marks will be assigned to them.) 

A. Show that if .4, B, Care the angles of a triangle, 

tan A + tan B + tan C = tan A tan B tan C. 
Show also that 

tanA5tan^C + tan*CtanA.i + tani^tan|^= 1. 

B. Solve the equation a cos d + h sin 6 = c. 

Find all the solutions of sin ^ sin 3 ^ = sin 5 6 sin 7 0. 



COLLEGE OF PRECEPTORS. 

Cheistmas, 1910. 

H Hours. 

[Four-place tables of logarithms and of natural functions and 
square-ruled paper are provided. All diagrams should be 
drawn as accurately as possible.] 

Part I. 

1. Define a radian, and find its magnitude in degrees to two 
places of decimals (tt = -->'). 

If the angle of an equilateral triangle were taken to be the unit 
angle, what would be the measure of a radian to two places of 
decimals ? 



196 EXAMINATION PAPERS 

2. Define the sine and tangent of an acute angle. Prove that 

sin'^^ + cos'^^ = 1. 
If tan^ = i\^, find the value of cos ^-8 sin ^. 

3. Find, geometrically, tan 30°. 

If ^ = 30°, B = 45°, C = 60°, D = 90°, find the value of: 
(i) sin ^ cos 5 — sin BcosA; 
(ii) (tan^^ - cosec^^) / (cot C + cos D). 

4. Use logarithms to find as nearly as possible the values of: 
(i) 3-142 X •9342/-00532 ; (ii) >/562'3/-00^984. 

5. Solve, using the tables, the triangle in which C = 90°, 
« = 654, ^ = 38°45'. 



Part II. 

6. Find all the positive values of 0, less than 360°, which satisfy 
the equations : 

(i) cos2^-sin"^ = 0; 
(ii) 4sin2^cos^^-sin^<9= |. 
Which of the following statements are possible ? 

(i) tan^= -2; (ii) sin^ = f. 

7. Write down, without proof, the expansions of sin (.1 - B), 
cos{A-B). 

Find the value of ianA — B in terms of tan J, tan 2?. 
If tan^l = ^, tan J5 = -/, find tan (.4 + ^). 

8. Prove that, in a triangle, a^ = &- + r-26fcos.4 when the 
angle A is (i) acute, (ii) obtuse. 

Deduce that tan lA = \/ (s -h) {s - c) -^ s [s - a). 
Find the greatest angle in the triangle whose sides arc 256, 
389, 401. 

9. AB is a horizontal straight line. A vertical straight line is 
drawn from B upwards, and in it two points P, Q are taken, such 
that BQ is five times BP. If the angle BAP is 30'\ calculate 
UmPAQ. 



EXAMINATION PAPKRS 107 

LEAVING CERTIFICATE EXAMINATION 
(SCOTLAND). 1910. 

1. Explain the circular measurement of angles. 
Express 30°, 50°, 166° 40' in radians. 

Express '0187 radian in degrees, minutes, and seconds, taking 
77 = 3-1416. 

2. Taking a horizontal inch to represent 10° and 5 vertical inches 
to represent the unit of length, plot, with the help of your tables, 
the values of tan 6 when ^ = 0, 10°, 20°, 30°, 40°, 50°. 

Plot also the values of sin 6 for the same angles, join both series 
of points by smooth curves, and thus find a graphic solution of the 
equation 5 (tan ^- sin ^) = 1. 

3. State the relation which exists between the sine and cosine of 
any angle. 

Use this relation to find, and express in a diagram, all the values 
of 0^, less than 180°, which satisfy the equation 
5 sin a + 6 cos'' 3^ = 7. 

Either, 4 a. A man walked 5 miles due North and then walked 
6 miles in a direction 27° East of North. Find by a figure drawn 
to scale how far he now is from his starting-point, and in what 
direction he should have originally started in order to go straight 
to his final position. Verify your results by calculation. 

Or, 4 b. The sides of a parallelogram are 2 inches and 3 inches 
in length, and its area is 3? square inches. Find by a diagram the 
sizes of its angles and the length of its longer diagonal. Verif}- 
your results by calculation. 

Either, 6 a. Draw a circle of radius 2 inches, and inscribe in it 
a triangle ABC, such that ZB = .34°, ZC= 73°. 

Measure the lengths of the sides as nearly as possible. 

Calculate with the help of the tables the lengths of the sides to 
the nearest hundredth of an inch, and thus test the correctness of 
your di awing. 

Or, 5 b. State and prove the formula which gives tan(yl + i?) in 
terms of tan A and tan B. 

Apply this formula to find expressions for tan 2^, tan3yl, and 
tan 5 ^ in terms of tan A. 



198 EXAMINATION PAPERS 

INTERMEDIATE EXAMINATION (IRELAND). 

Middle Grade (Pass). 1910. 

1. Prove that sin^^ + cos^^ = 1, where A is an obtuse angle. 

2. Find the value of the expression cosec^-gcot^, if sinyl = ^^, 
when A is acute, and when A is obtuse. 

3. Prove the identity (1 - tanM) -r- (2 cosM - 1) = secM. 

4. In a triangle C = 90°, c = 65°, tan A = '28. Find a and h 
each to two decimal places. 

5. In a triangle a = b \/3, /; = 11, C = 150°. Find c and cos^. 

6. In a triangle B = 45°, fc = 20, c = 4. Find sin C, and prove 
that the perpendicular from A on BC divides BC into two 
segments one of which is seven times the other. 

7. Prove that the length of the perpendicular from the vertex A 
of a triangle on the opposite side BC is equal to «/(cot-B + cotC), 
considering the cases when both angles^ are acute, when one is 
right, and when one is obtuse. 

8. Find the angles between 0° and 360° which satisfy the equation 
6sin^-4 cosec^ + cot6^ = 0, being given cos 48° IT 23" = §. 

Middle Grade (Honours). 1910. 

1. Show by a graph the values of cosec A for values of A between 
-90° and 360°. 

2. If yl is an angle in the first quadrant, prove that 

sin A + cos A 4- tan A + cot ^ > sec ^ + cosec A. 

3. Prove the identity 

3 (sin ^ - cos ^)H (sin A + cos Af + 4 (sinM + cos^i) = 13. 

4. The sides of a triangle are 37, 7, and 40. Find all the angles, 
being given that eo? 69° 25' 48" = ?; ; . 

5. In a triangle a = \/5, h = -y/lS, C = 45°. Find c, and prove 
that cotyl = 2-v/f-l. 

6. Find a solution between 180° and 270° of the equation 

5(1+ sin .r) = — 3 cos .r, 
being given cos 28° 4' 21" = 1;. 



EXAMINATION PAPEES 199 

7. Prove by drawing a line through B, making an angle x with 
the side BC, or otherwifse, that in a triangle ABC, 
c cos {B - x) + h COS {C + x) = acosx. 

P is a point on the hypotenuse AB of a right-angled triangle 
ABC. AP = x, PB = ij, PC = z. Find cos CPB interms of jr, y, 
and z. Find the sides of the triangle when x = S — ^/d, y = ^/3 + 1 , 

z= ye. 

Senior Grade (Pass). 1910. 

1. Find the distance from the earth to the moon, assuming that 
the moon's diameter, 2165 miles, subtends an angle of 31' lU" at 
the earth. 

2. Prove that tan \A = {\- cos ^)/sin A. 

Find tan 15° and tan 22i" without using the tables. 

3. Find x if cos-^;r-f cot"' 2 = ^tt. 

4. Assuming the formulae for the sines of the sum and difference 
of two angles, prove that 

sin A-sinB = 2cosl{A + B] sin \ {A - B). 
Find the corresponding expressions in factors for cos^ -cos B. 

5. Find the solutions between 0' and 360^ of the equation 

cot 2. r- 3 tan re = 3. 

6. In a triangle a = 183, h = 247, C = Sr 40'. Find A and B. 

7. In a triangle A ^ 54° 80', B = 69° 20', a = 341. Find h and c. 

8. Prove that in a triangle 

a cos B-bcosA= {a-- h'^)/c. 

9. Prove that in a triangle rcot|^=s — a, where r is the 
radius of the inscribed circle, and s the perimeter. 

Senior Grade (Honours). 1910. 

1. An arc 40 feet in length is taken on a circle whose radius 
is 35 feet. Find, to the nearest inch, the length of the perpen- 
dicular from the centre on the chord of this arc. 

2. Prove the identity 

cos5^/sin^ + sin5^/cos^ = 2 cosec2^-4 sin 2 .4. 

3. If cos x + cos y + cos z + cos x cos y cos 2- = 0, prove that 

tan I X tan J y tan Iz = ± 1 . 

4. If X = cot~^ V'cos y — tan~' -y/cosy, prove that 

y = 2tan~^ -y/sin.r. 

5. In a triangle A = 35° 20', a = 127, h = 104. Fin<l B, C, 
and c. 



INDEX 



INDEX 



A.bscis.sa, 12. 
Addition formulae, 155, 
Altitude of sun, 89. 
Ambiguous case, 120. 
Angle, 25. 

circular measure of, 29. 

measurement of, 25. 

negative, 26. 
.»^f elevation, 88. 
>of depression, 88. 
Antilogarithm, 21. 
Arc, length of, 28. 

functions of, 38. 
Area of triangle, 109. 
Axes, 12. 

Characteristic, 18. 
Chord of angle, 15. 

ircular functi 

measure, 29. 
Complementary angle, ratios of, 

40. 
Co-ordinates, 12. 

cartesian, 12. 

polar, 41. 
Cosecant, 38. 
Cosine, 38. 

formula, 108, 

cos (A±B), 155. 

cos A ± cos B, 159. 
Cotangent, 38. 
Course, 94. 
Coversine, 40. 

Degree, 25. 

De Moivre's theorem, 152. 
Departure, 94. 
Difference of Latitude, 94. 

of two angles, ratios of, 15G. 

of two sines, 159. 

of two cosines, 159. 
Dip of the horizon, 127. 
Double angle, ratios of, 100, 162. 

Elementary formulae, 49. 
Eliminant, 51. 
Elimination, 51. 



Equations, solution of, 78. 
solution by graphs, 17, 76. 
general solution of, 79. 

Functions, algebraical, 14. 
circular, 40, 
trigonometrical, 40. 

Geometry enunciations, 9. 

Grade, 27. 

Gradient, 33. 

Graphs, general treatment, 12. 

of trigonometrical functions, 41, 
163. 

harder trigonometrical, 74. 
Greek alphabet, 24. 

Half-angle, ratios of, 101, 104, 121, 

162. 
Haversine, 40, 121. 
Heights and distances, 88, 124. 

-identities, 50, 64, 167, 168, 170. 
Imaginary quantities, 149. 
Indices, fractional, 17. 

negative, 17. 
Interpolation, 37. 
Inverse functions, 45, 164. 

Latitude, 32. 
Logarithm, 18. 

base of, 18. 

use of, 22, 81. 

tabular, 36. 
Longitude, 32. 

Mantissa, 19. 
Mariner's compass, 31. 



Minimum value, 74. 

Navigation : 

plane sailing, 94. 
parallel sailing, 95, 
middle latitude sailing, 96, 
traverse sailing, 96. 



204 



INDEX 



Negative angle, 26. 
ratios of, 54. 
direction, 13. 

Oral examples, 34, Gl, 116, 123, 

166. 
Ordinate, 12. 
Origin, 12. 



Point, 32. 

Points of the compass, 31. 

Powers of trigonometrical ratios, 

44. 
Product of sines and cosines, 160. 
Projection, definition, 139. 

of area, 147. 

propositions, 140. 

length of, 40. 

formulae, 40, 109. 
Proportional parts, 71. 

Quadrilateral, area of cyclic, 111. 

Radian, 29. 

Radius of inscribed circle, 114,168. 

of circumcircle, 114. 

of escribed circle, 114, 168. 
Rate of change of function, 76. 
Ratios, trigonometrical, 37. 

of 90-^, 53. 

of 180-^, 53. 

of -^,54. 

of 0° and 90^, 56. 

of 30°, 45°, 60°, 57. 

of 2 A, 100, 162. 

of i A, 101, 162. 

of 3 A, 164. 

ofA + B, 154. 

of A -B, 156. 

ofSA, 164. 
Rhumb line, 94. 



Secant, 38. 
Sector, area of, 29. 
Sine, 38. 

formula, 107. 

graph, 41. 

sin {A±B), 154. 

sin A ± sin B, 159. 
Slope of curve, 76. 
Small angle, 59. 

sine and tangent of, 60. 
>* Solution of right-angled triangle, 
88. 

of triangles, 119. 

of equations, 78, 168. 

of equations by graphs, 17, 76. 
Sum of two angles, ratios of, 
154. 

of two sines, 159. 

of two cosines, 159. 
Summation of a series, 169, 
Supplementary angle, ratios of, 

53. 



Tabular logarithm, 36. 
Tangent, 38. 

graph, 41. 

tan iA±B), 163. 
Triangle formulae, 107. 

solution of, 119. 
Trigonometrical ratios : 

general definition of, 38. 

right-angled triangle, definition 
of, 86. 
Turning points, 74. 



Vectors, 143. 

addition of, 144. 

resolution 
Versine, 40. 



Five-figure Logarithmic 

and 

Trigonometrical Tables 

ARRANGED BV 

W. E. PATERSON, M.A., B.Sc. 

MATHEMATICAL MASTER, MERCERs' SCHOOL 
AUTHOR OF 'SCHOOL ALGEBRa/ 'ELEMENTARY TRIGONO.METRY ' 



OXFORD: AT THE CLARENDON PRESS 
LONDON: HENRY FROWDE, AMEN CORNER, E.C. 

AXD AT 

EDINBURGH, GLASGOW, NEW YORK, TORONTO 
AND MELBOURNE 



OXFORD : HOKACE HAET 
TRINTER TO THE UNIVERSITY 



These five-figure tables are intended to give results 
correct to four figures ; tlie fifth figure in tlie answer may 
be inaccurate. 

The decimal point is printed before all the logarithms 
of numbers ; it is hoped that this will obviate the common 
mistake of reading off logarithms instead of antilogarithms, 
and vice -versa. 

The trigonometrical tables are arranged so that, at one 
opening of the tables, all the functions of an angle may 
be found on the left-hand page and their logarithms on 
the right-hand page ; here again confusion is avoided. 
The characteristics of the logarithmic functions are the 
true characteristics ; no useful purpose is served by 
increasing them by 10. 

It should be noticed that, instead of dividing by a sine, 
one may multiply by the cosecant, &c., and, similarly, 
instead of subtracting the logarithm of a sine, one may add 
the logarithm of the cosecant, &c. In many cases this 
shortens calculation. 

For quick reference the last page may be used, which 
gives the trigonometrical functioas, to four figures only, 
for every whole degree up to 90° and the corresponding 
circular measure to five figures. 



R 

1-0025 
1-005 
1-0075 
i-oi . 
1-0125 
I -015 
I -0175 
I -02 
1-0225 
1-025 
1-0275 
I -03 . 



Logarithms of R for 

log R 

00108438 
00216606 
00324505 

00432137 
00539503 
00646604 
00753442 
00860017 
00966332 
01072387 
01178183 
01283722 



Compound Interest 

R logR 

1-0325 . , . . . -01389006 

I -035 -01494035 

I-0375 -01598811 

I-04 . . ' . . . . -01703334 

1-0425 -01807606 

1-045 •01911629 

I -0475 -02015403 

1-05 . . . . . , -02118930 

1-0525 -02222210 

1-055 -02325246 

1-0575 -02428038 

I -06 -02530587 



Constants used in Mensuration and their Logarithms 



7r= 3-14159265 
in = 1-57079633 
iyr =052359878 

•A 77 = 4-18879020 

^/n= 1-77245385 

7f2 = 9-86960440 

^n= I -46459 1 89 
7r/l8o = 0-01745329 



logarithm 
0-497150 
0-196120 
1-718999 
0-622089 
0-248575 
0-994300 
0-165717 
2-241877 



I -=- 77 
1 -^ 477 

-</ 6-r~ ^ 

\/ 3 -^ 4 TT 

\/l 4- 77 



= 0-31830989 
= 0-07957747 
= 1-24070098 
= 0-62035049 
= 0-56418958 



1 -^ 77^= 0-10132118 
^^= 2-14502940 
180/77= 57-29577951 



logarithm 
1-502850 
2-900790 
0-093667 
1-792637 

1751425 
1-005700 

0-331433 
1-758123 



Naperian (or Natural) Logarithms 
e= 2-7182182 logjoe = -43429448 logg 10 = 2-30258509 
logio -V = logg A" X logio e. log^ A' = logio ^V x logg 10 



LOGARITHMS OF NUMBERS 



Mean Differences 



^1 



10 

II 

12 
13 

14 

IS 

j'i) i6 

7-^_i7 
i8 

J9 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
34 
35 
36 

37 
38 

39 



40 
41 
42 

43 
44 
45 
46 
47 
48 
49 

50 
51 
52 
S3 

54 





•00000 

•04139 

•07918 

•II394 
•I46I3 
P7609 
•20412 

•23045 
•25527 
•27875 

•30103 

32222 

■34242 

36173 
38021 

•39794 

•41497 

•43136 
•44716 

•46240 

•47712 

•49136 

•50515 
•5I85I 

•53148 
S4407 

55630 
56820 
57978 
59106 



1 



•00432 

•04532 

•08279 

•II727 

1-14922 

•17898 

•20683 

•23300 
•25768 
•28103 



•60206 

•61278 
•62325 

■^^3347 
■64345 
65321 
66276 
67210 
68124 
69020 



•30320 

•32428 

•34439 
•36361 

•38202 

•39967 

•41664 

•43297 
•44871 
•46389 

•47857 

•49276 
•50651 
•51983 
•53275 
•54531 
'55751 
56937 
58093 
59218 



l 



69897 

•70757 
71600 

72428 I -72509 
3239 I -73320 



•60314 

•61384 
•62428 
•63448 

•64444 
•65418 

•66370 
•67302 
•6S215 
•69108 

69984 

■70S42 
■71684 



•00860 

-04922 
-08636 
•12057 
•15229 
•18184 
20952 

'23S53 
•26007 
•28330 

•30535 

•32634 

•34635 
•36549 
'3^3^2 
•40140 
•41830 

•43457 
•45025 

•46538 

•48001 

•49415 
•50786 

•52114 
•53403 
•S4654 

•55871 
•57054* 
•58206 

•59329 



3 

•01284 

•05308 
0899 r 
•12385 

•15534 
-18469 

-21219 
-23805 
•26245 
-28556 

•30750 

-32838 
•34830 
■36736 
•38561 
•40312 
■41996 
■43616 

45179 
46687 



•01703 

•05690 
-09342 
•12710 
•15836 
•18752 
-21484 

•24055 
-26482 
•28780 

•30963 

•33041 
•35025 
•36922 
•38739 
40483 
42160 
43775 
45332 
46835 



•60423 

•61490 
•62531 
•63548 
•64542 
•65514 
■66464 

■67394 
68305 
69197 



•48144 

•49554 
•50920 
•52244 
■53529 
54777 
55991 
57171. 
58320 

59439 



60531 

•61595 
•62634 
•63649 
•64640 
•65610 
•66558 
•67486 
■68395 
■69285 



•70070 -70157 70243 

-70927 -71012 •71096 
•71767 ^71850 ^71933 
•72591 •72673-72754 
-73400 (-7^480 1 -731; 60 



•48287 

•49693 
•51055 

•52375 
•53656 
•54900 

•56110 
•57287 
•58433 
•59550 

60638 

61700 
•62737 

•63749 
•64738 
•65706 
•66652 
•67578 
-68485 
•69373 



42 

39 
35 
33 
•o 
28 
27 
25 
24 
23 

21 

20 

19 
18 

17 

17 
16 

15 

15 



14 

14 
13 
13 
13 

12 

12 
12 
II 
II 

II 

I I 
10 
10 
TO 
10 

9 
9 
9 
9 

9 

8 

8 
8 



85 

77 
71 
66 
61 
57 
S3 
50 
48 
45 

43 

41 
39 
37 
36 
34 
33 
32 
31 
30 



127 

116 
106 
98 
91 
85 
80 
76 
71 
68 

64 

6 

59 
56 
54 
52 

50 
48 
46 

45 



170 

155 
142 

131 
122 
114 

107 

lOI 

95 
90 

86 

82 
7^ 

75 
72 

69 

66 
64 
61 
59 



43 57 



33 44 



32 

32 
31 
30 
29 
29 
28 
28 

27 
26 

26 

25 

25 



43 

42 
41 
40 
39 
38 
38 
37 
36 
35 

35 

34 
33 



212 

193 
177 
164 
152 
142 

134 
126 
119 
113 

107 

102 
98 
94 
90 
86 

S3 
80 

77 
74 

72 

70 
67 
65 
63 
62 
60 
58 
57 
55 



254 

232 
213 
197 
183 
171 
160 
151 

'•$^ 
i'35 



297 

270 
248 
229 

213 
199 

187 
176 
167 

i^r 



1281150 

1231143 

ii7|i37 
112 131 



108 
103 

99 
96 
92 
89 



8 16 



24 33 

24 1 32 



54 

53 
51 
50 
49 
48 
47 
46 

45 
44 

43 

42 
42 



S3 
81 

78 
76 
74 
72 
70 
68 
66 

65 

63 
62 
60 
59 
58 
56 
55 
54 
53 

52 

51 
50 



125 
120 

116 
112 
108 
104 



339 

309 
284 
262 

244 
228 

21,4 
201 
190 
180 



86 10 1 



411 49 
40 



97 
94 
91 
89 
86 
84 
82 
80 
78 

76 

74 
72 
70 
69 
67 
66 
64 

63 
62 

60 

59 
58 
57 



48 I 56 



172 

164 
158 
149 

143 
138 

132 
128 
123 
119 

IIS 

111 
108 
105 

101 

99 

96 

93 
91 
89 

86 

84 
82 
80 
79 
77 
75 
73 
72 
71 



31 

3^ 
3' 

2C 
2; 
21 

24 
22 
21 
2C 

19 

18 

16 
16: 
15' 

14. 

14. 
13; 
13^ 

I2( 

I. 

121 

11^ 

114 

II] 

108 

105 

102 

100; 

57 

95 
93 
90 
88 
86 
84 

83 
81 

79 



69 78 



68 
67 
65 
64 



76 

75 

73-^ 

72 



LOGARITHMS OF NUMBERS 





5 


6 


7 


8 


9 


I 2 


^ 


4 


' 


6 


7 


8 


9 


lO 


02119 02531 


•02938 


03342 


•03743 


40 sT 


121 


162 


202 


242 


283 


323 


364 


II 


•06070 -06446 


-06819 


-07188 


•07555 


37|74 


III 


148 


185 


222 


259 


296 


III 


12 


•09691 -10037 


-10380 


-10721 


•11059 


34|68 


102 


^37 


170 


204 


238 


272 


307 


13 


•13033 -13354 


m 


-13988 


-14301 


32:63 


i 


126 


1^8 
147 


190 


221 


253 


284 


14 


•16137 -16435 


-17026 


•17319 


29 59 


118 


177 


206 


236 


265 


15 


•19033 19312 


•19590 


•19866 


•20140 


28 55 


83 


no 


138 


165 


193 


221 


248 


i6 


•217481 -22011 


■22272 


•22531 


-227S9 


26 52 


is 


104 


130 


156 


182 


208 


233 


17 


•24304-24551 


•24797 


•25042 


•25285 


24 49 


73 


98 


123 


147 


171 


196 


220 


i8 


•26717! -269^1 


•27184 


•27416 


•27646 


23 


46 


70 


93 


116 


139 


162 


185 


208 


19 


•29003 


-29236- 


"•29447 


•29667 


-29885 


22 


44 


66 


88 


no 


132 


154 


176 


198 


20 


•31175 


•31387 


•31597 


-31806 


•32015 


21 


42 


63 


84 


los 


126 


147 


168 


188 


21 


•33^44 


•33445 


•33646 


•33846" 


•34044 


20 40 


60 


80 


100 


120 


140 


160 


180 


22 


■35218 -35411 


•35603 


•35793 


•35984 


19 38 


57 


77 


96 


H15 


134 


153 


172 


23 


■37107 -37291 


•37475 


•37658 


•37840 


18,37 


55 


73 


91 


no 


128 


146 


165 


24 


•38917 -39094 


-39270 


•39445 


•39620 


i8!35 


53 


70 


88 


105 


123 


140 


158 


25 


40654 1 -40824 


•40993 


•41 162 


•41330 


17 34 


51 


67 


84 


lOI 


118 


135 


152 


26 


•423251-42488 


-42651 


-42813 


•42975 


16 I 32 


49 


65 


81 


97 


114 


130 


146 


27 


•43933 


•44091 


-44248 


•44404 


-44560 


16 31 


47 


63 


78 


94 


no 


125 


141 


28 


•45484 


•45637 


-45788 


•45939 


-46090 


i5j3o 


45 


61 


76 


91 


io6 


121 


136 


29 


•46982 


•47129 


•47276 


•47422 


•47567 


14; 29 

1 


44 


58 


73 


87 


102 


117 


131 


30 


48430 


•48572 


•48714 


-48855 


•48996 


1 ^ 
14 28 


42 


56 


71 


85 


99 


113 


127 


31 


■49831 P49969 


-50106 


-50243-50379 


14 27 


41 


55 


68 


82 


96 


109 


^23 


32 


•51188 -51322 


•51455 


-51587 -51720 


^1,27 


40 


53 


66 


80 


93 


106 


119 


33 


•52504 -52634 


■52763 


-52892 -53020 


13 26 


39 


51 


64 


77 


90 


103 


116 


34 


•53782! -53908 .54033 


-54158 1-54283 


13 25 


3^ 


50 


63 


75 


88 


100 


113 


35 


•S5023 1 55145 


•55267 55388 -55509 


12 24 


36 


49 


61 


73 


85 


97 


109 


36 


■56229 1-56348 


•56467 1 -56585 1-56703 


12 24 


31 


47 


59 


71 


S3 


95 


106 


37 


■57403. 


•57519 


•57634 


•57749 


•57864 


12,23 


35 


46 


58 


69 


81 


92 


104 


38 


•58546 


•58659 


•58771 


•58883 


•58995 


11,22 


34 


45 


56 


67 


78 


90 


lOI 


39 


•596601-59770 


•59879 


•59988 


-60097 


II 22 


33 


44 


55 


66 


76 


87 


98 


40 


1 1 

•60746 -60853 ! -609591 -61066 


-61172 


II 21 


32 


43 


53 


64 


74 


8S 


96 


41 


•61805 -^'909 


-62014 •62118 


•62221 


10 21 


31 


42 


52 


62 


73 


83 


94 


42 


•62839 


-62941 


•63043 ^63144 


•63246 


10 20 


30 


41 


51 


61 


71 


81 


91 


43 


•63849 


-63949 


•64048 -64147 j -64246 


10 20 


30 


40 


50 


60 


70 


79 


89 


44 


•64836 


•64933 


•65031 -65128-65225 


10 19 


29 


39 


49 


58 


68 


78 


87 


45 


65801 


-65986 


•65992 66087 66181 


10 19 


29 


38 


48 


57 


67 


76 


86 


46 


•66745 


-66839 


-66932 


-670251-67117 


9 19 


28 


37 


47 


56 


65 


74 


84 


47 


•67669 


•67761 


-67852 


•67943 


-68034 


9! 18 


27 


36 


46 


55 


64 


73 


82 


48 


•68574 


•68664 


•68753 


-68842 


•68931 


9!i8 


27 


36 


45 


54 


62 


71 


80 


49 


•69461 


-69548 


-69636 


-69723 


-69810 


9;i7 

1 


26 


35 


44 


52 


61 


70 


78 


SO 


70329 


•70415 


•70501 


•70586 ^70672 


9 17 


26 


34 


43 


SI 


60 


68 


77 


51 


■71181 


-71265 


•71349 


•7i433i^7i5i7 


8,1/ 


25 


34 


42 


50 


59 


<^7 


75 


52 


•72016 


-72099 


-72181 


■72263 


•72346 


8 16 


■^5 


33 


41 


49 


58 


66 


74 


S3 


•72835 


•72916 


-72997 


•7307s 


•73159 


8!i6 


24 


12 


40 


48 


57 


65 


73 


54 


-73640 


•73719 


•73799 


■73878 


•73957 


8|i6 


24 


32 


40 


48 


55 


63 


71 



Mean Differences 



LOGARITHMS OF NUMBERS 



Mean Differences 





1 1 1 


2 


3 


4 


I 


2 


3 


4 


5 


6 


7 


8 


9 


55 


•74036 


•741 15 


•74194 


•74273 


•74351 


8 


^ 


24 


3! 


39 


47 


55 


6j 


71^ 


56 


74819 


•74896 


•74974 


75051 


•75128 


8 


15 


^l 


31 


39 


46 


54 


62 


69 


57 


75587 


75664 


•75740 


75815 


75891 


8 


15 


23 


30 


38 


46 


53 


61 


68 


58 


76343 


•76418 


•76492 


•76567 


•76641 


7 


15 


22 


30 


1>7 


45 


52 


60 


67 


59 


77085 


•77150 


77232 


•77305 


'77119 


7 


15 


22 


29 


17 


44 


51 


59 


66 


6o 


•77?i5 


•77887 


•77960 


•78032 


•78104 


7 


14 


22 


29 


36 


43 


51 


58 


65 


6i 


78533 


•78604 


•78675 


•78746 


78817 


7 


14 


21 


28 


36 


43 


50 


57 


64 


62 


79239 


•79309 


•79379 


•79449 


•79518 


7 


14 


21 


28 


35 


42 


49 


56 


63 


63 


79934 


•80003 


•80072 


•80140 


•80209 


7 


14 


21 


27 


34 


41 


48 


55 


62 


64 


•80618 


•80686 


•80754 


•80821 


•80889 


7 


14 


20 


27 


34 


41 


47 


54 


61 


65 


•81291 


•81358 


•81425 


•8149 1 


•81558 


7 


13 


20 


27 


33 


40 


47 


53 


60 


66 


•81954 


•82020 


•82086 


•82151 


•82217 


7 


13 


20 


26 


Zl 


39 


46 


52 


59 


67 


•82607 


•82672 


'^^7Z7 


•82802 


•82866 


6 


13 


19 


26 


32 


39 


45 


52 


58 


68 


•83251 


•83315 


•83378 


•83442 


•83506 


6 


13 


19 


25 


Z^ 


Z^ 


45 


51 


57 


69 


•83885 


•83948 


•8401 1 


•84073 


•84136 


6 


13 


19 


25 


31 


38 


44 


50 


56 


70 


•84510 


•84572 


•84634 


•84696 


•84757 


6 


12 


19 


25 


31 


37 


43 


49 


S6 


71 


•85126 


•85187 


•85248 


•85309 


•85370 


6 


12 


18 


24 


31 


17 


43 


49 


55 


72 


•85733 


•85794 


•85854 


•85914 


•85974 


6 


12 


18 


24 


30 


36 


42 


48 


54 


73 


•86332 


•86392 


•86451 


•86510 


•86570 


6 


12 


18 


24 


30 


36 


42 


48 


53 


74 


•86923 


•86982 


•87040 


•87099 


•87157 


6 


12 


18 


23 


29 


35 


41 


47 


53 


75 


•87506 


•87564 


•87622 


•87679 


•87737 


6 


II 


17 


23 


29 


35 


40 


46 


52 


76 


•88081 


•88138 


•88195 


•88252 


•88309 


6 


II 


17 


23 


29 


34 


40 


46 


51 


77 


•88649 


•88705 


•88762 


•88818 


•88874 


6 


II 


17 


22 


28 


34 


39 


45 


51 


78 


•89209 


•89265 


•89321 


•89376 


•89432 


6 


II 


17 


22 


2\ 


ZZ 


39 


44 


50 


79 


•89763 


•89818 


•89873 


•89927 


•89982 


5 


II 


16 


22 


27 


Zl 


38 


44 


49 


80 


•90309 


•90363 


•90417 


•90472 


•90526 


5 


II 


16 


22 


27 


33 


38 


43 


49 


81 


•90849 


•90902 


•90956 


•91009 


•91062 




1 1 


16 


21 


27 


32 


Z7 


43 


48 


82 


•91381 


•91434 


•91487 


•91540 


•91593 




II 


16 


21 


27 


Z^ 


Z7 


42 


48 


83 


•91908 


•91960 


•92012 


•92065 


•92117 




10 


16 


21 


26 


31 


Z7 


42 


47 


84 


•92428 


•92480 


•92531 


•92583 


•92634 




10 


15 


21 


26 


31 


36 


41 


46 


85 


•92942 


•92993 


•93044 


•9309s 


•93146 




10 


15 


20 


26 


31 


36 


41 


46 


86 


•93450 


•93500 


•93551 


•93601 


•93651 




10 


15 


20 


25 


30 


35 


40 


45 


87 


•93952 


•94002 


•94052 


•94101 


•941 5 1 




10 


15 


20 


25 


30 


35 


40 


45 


88 


•94448 


•94498 


•94547 


•94596 


•94645 




10 


15 


20 


25 


30 


34 


39 


44 


89 


•94939 


•94988 


•95036 


•95085 


•95134 




10 


15 


19 


24 


29 


34 


39 


44 


90 


•95424 


•95472 


•95521 


•95569 


95617 




10 


14 


19 


24 


29 


34 


39 


43 


91 


•95904 


•95952 


•99599 


•96047 


•96095 




10 


14 


19 


24 


29 


ZZ 


Z'^ 


43 


92 


•96379 


•96426 


•96473 


•96520 


•96 5 Cy 
•970/5 




9 


14 


1 9-. 


24 


2% 


zz 


38 


42 


93 


•96H48 


•96895 


•96942 


•96988 




9 


14 


19 


23 


28 


ZZ 


Z7 


42 


94 


•97313 


•97359 


•97405 


•97451 


•97497 




9 


14 


18 


23 


2"^ 


z- 


Z7 


41 


95 


•97772 


•97818 


•97864 


•97909 


•97955 




9 


14 


18 


23 


27 


32 


36 


41 


96 


•98227 


•98272 


•98318 


•983^>3 


•9S408 




9 


14 


18 


^}< 


-7 


32 


36 


41 


97 


•9H677 


•98722 


•98767 


•9881 1 


•98856 




9 


13 


18 


Zl 


^7 


31 


36 


40 


98 


.99123 


•99167 


•992 1 1 


•99255 


•99300 




9 


13 


18 


22 


^7 


31 


35 


40 


.99 


•99564 


•99607 


•9965 1 


•99695 


•99739 


4 


9 


13 


17 


22 


26 


31 


35 


39 













LOGARITHMS OF NUMBERS 




5 6 7 


8 1 9 


I 

8 


2 
16 


3 
23 


4 5 
3TI39 


1'. 
47 


7 
55 


8 
62 


9 


55 


•74429 -74507 •74586 


•74663 


-74741 


70 


56 


•75205 


•75282 1 -75358 


•75435 


-75511 


8 


15 


23 


31 


38 


46 


53 


61 


69 


57 


•75967 


•76042. -76118 


•76193 


•76268 


8 


15 


23 


30 


38 


45 


53 


60 


68 


58 


76716 


•767901-76864 


•76938 


-77012 


7 


15 


22 


30 


37 


44 


52 


59 


66 


59 


•77452 


•77525 -77597 


'77^70 


'777AZ 


7 


15 


22 


29 


36 


44 


51 


58 


65 


6o 


•78176 


•78247 78319 


•78390 


•78462 


7 


14 


21 


29 


36 


43 


50 


57 


64 


6i 


•ySiSS 


•78958^79029 


-79099 


•79169 


7 


14 


21 


28 


35 


42 


49 


56 


63 


62 


•79588 


•79657 -72Z27 


-79796 


•79865 


7 


14 


21 


28 


35 


42 


48 


55 


62 


. 63 


•^ozjj j -803461 -So^l^^ 


•80482 


•80550 


7 


14 


20 


27 


34 


41 


48 


55 


61 


64 


•80956! -81023 -81090 


•81158 


•81224 


7 


13 


20 


27 


34 


40 


47 


54 


60 


65 


•81624 81690 81757 


•81823 


•81889 


7 


13 


20 


26 


33 


40 


46 


53 


59 


66 


•82282 


•82347 ,•82413 


•82478 


-82543 


7 


13 


20 


26 


33 


39 


46 


52 


59 


67 


•82930 


-82995 


-83059 


-83123 


-831^ 


6 


13 


19 


26 


32 


39 


45 


51 


58 


68 


•83569 


-83632 


•83696 


-83759 


•83822 


6 


13 


19 


25I32 


38 


44 


51 


57 


69 


•84198 


•84261 


■^AZ27, 


-84386 


•84448 


6 


12 


19 


25 


31 


37 


44 


50 


56 


70 


•S4819 


•84880 


-84942 


-85003 


•85065 


6 


12 


18 


25 


31 


37 


43 


49 


55 


71 


•^'5431 


•85491 


-85552 


•85612 


•85673 


6 


12 


18 


24 


30 


36 


42 


48 


54 


72 


•^6034 


•86094 


-86153 


•86213 


■m27z 


6 


12 


18 


24 


30 


36 


42 


48 


54 


73 


•86629 


•86688 


-86747 


-86806 


•86864 


6 


12 


18 


24 


29 


35 


41 


47 


53 


74 


•87216 


-87274 


■^73i2 


-87390-87448! 


6 


12 


17 


23 


29 


35 


41 


46 


52 


75 


•87795 


•87852 


•87910 


•87967 


-88024 


6 


II 


17 


23 


29 


34 


40 


46 


51 


76 


•88366 


•^H2Z 


•88480 


-88536 


•88593 


6 


II 


17 


23 


28 


34 


40 


45 


51 


77 


•88930 


•88986 


•89042 


•89098 


•89154 


6 


II 


17 


22 


28 


33 


39 


45 


50 


78 


•89487 


-89542 


-89597 


•■89653 


•89708 


6 


II 


17 


22 


28 


33 


39 


44 


50 


79 


■90037 


•90091 


•90146 


•90200 


•90255 


5 


II 


16 


22 


27 


33 


38 


44 


49 


80 


•90580 


-90634 


•90687 


•90741 


^90795 


s 


II 


16 


22 


27 


32 


38 


43 


48 


81 


•91 1 16 


•9II69 


•91222 


•91275 


•91328 


5 


II 


16 


21 


27 


32 


37 


42 


48 


82 


•91646 


•91698 


•91751 


-91803 


•91855 


5 


10 


16 


21 


26 


31 


37 


42 


47 


83 


•92169 


-92221 


.92273 


•92324 


•92376 


5 


10 


16 


21 


26 


31 


36 


41 


47 


84 


•92686 


•92737 


-92788 


•92840 


•92891 


5 


10 


15 


20 


26 


31 


36 


41 


46 


85 


•93197 93247 


-93298 


-93349 ! 93399 


5 


10 


15 


20 


25 


30 


35 


40 


46 


86 


•93702-93752 


•93802 


•93852 ^93902 


5 


10 


15 


20 


25 


30 


35 


40 


45 


87 


•94201 1-94250 


•94300 


•94349 


•94399 


5 


10 


15 


20 


25 


30 


35 


40 


44 


8a 


:24694 


•94743 


-94792 


•94841 


•94890 


5 


10 


15 


20 


25 


29 


34 


39 


44 


89 


•95182 


•95231 


•95279 


•95328 


-95376 


5 


10 


15 


19 


24 


29 


34 


39 


44 


90 


•9566s 


•95713 


-95761 


-95809 


•95856 


5 


10 


14 


19 


24 


29 


33 


38 


43 


91 


•96142 


•96190 


-96237 


•96284 


-96332 


5 


9 


14 


19 24 


28 


33 


38 


43- 


92 


•96614 


-96661 


-£6708 


-96755 


-96802 


5 


9 


14 


19 


23 


2S 


33 


37 


42 


93 


•97081 


•97128 


•97174 


•97220 


-97267 


5 


9 


14 


19 


23 


28 


32 


37 


42 


94 


•975431-97589 


•97635 


•97681 


.97727 


5 


9 


14 


18 


23 


27 


32 


37 


41 


95 


•98000 98046 98091 


•98137 i 98182 


5 


9 


14 


18 


23 


27 


32 


36 


41 


96 


•98453-98498 -98543 


•98588 


•98632 


4 


9 


13 


18 


22 


27 


31 


36 


40 


97 


•98900 -98945 


-98989 


-99034 


•99078 


4 


9 


13 


18 


22 


27 


31 


36 


40 


98 


•99344, -99388 


•99432 -99476 


•99520 


4 


9 


13 


18 


22 


26 


31 


35 


40 


J 99_ 


•99782 -99826 


-99870 '-99913 


•99957 


4 


9 


13 


17 


22 


26 


31 


3<> 


39 



Mean Differences 



ANTILOGARITHMS 



Mean Differences 








1 


2 


3 


4 


2 
2 5 


3 
~7 


4 
~9 


5 
12 


6 


7 
i6 


8 
19 


9 


•oo 


1 0000 


10023 


10046 


10069 


10093 - 


21 


•01 


10233 


10257 


10280 


10304 


10328 2 5 


7 


9 


12 


14 


17 


19 


21 


•02 


1047 1 


10495 


10520 


10544 


10568 2 5 


7 


10 


12 


15 


U 


20 


22 


•03 


10715 


10740 


10765 


10789 


10814 2 5 


7 


10 


12 


15 


17 


20 


22 


•04 


10965 


10990 


1 1015 


11041 


1 1066 3 5 


8 


10 


13 


15 


18 


20 


23 


•OS 


11220 


1 1 246 


1 1272 


1 1298 


I 1324 : 


J 5 


8 


10 


13 


16 


18 


21 


23 


•06 


1 1482 


1 1 508 


"535 


11561 


11588 3| 5 


8 


II 


13 


16 


18 


21 


24 


•07 


1 1 749 


1 1776 


1 1803 


11830 


11858 : 


5 5 


8 


II 


14 


16 


19 


22 


24 


•08 


12023 


12050 


12078 


12106 


12134 . 


5 6 


8 


11 


14 


U 


19 


22 


25 


•09 


12303 


12331 


12359 


12388 


12417 . 


5 6 


9 


11 


14 


17 


20 


23 


26 


•10 


12589 


12618 


12647 


12677 


12706 i 


( 6 


9 


12 


15 


18 


20 


23 


26 


•II 


12882 


12912 


12942 


12972 


13002 : 


5 6 


9 


12 


15 


18 


20 


24 


27 


•12 


13183 


13213 


13243 


13274 


13305 : 


5 6 


9 


12 


15 


18 


21 


24 


27 


•13 


13490 


13521 


13552 


13583 


1 3614 : 


6 


9 


12 


16 


19 


22 


25 


28 


•14 


13804 


13836 


13868 


13900 


13932 : 


6 


10 


13 


16 


19 


22 


26 


29 


•15 


14125 


141S8 


14191 


14223 


14256 3 


i 7 


10 


13 


16 


20 


23 


26 


30 


•16 


14454 


14488 


14521 


14555 


14588 : 


7 


10 


13 


17 


20 


24 


27 


30 


•17 


14791 


14825 


14859 


14894 


14928 : 


7 


10 


14 


17 


21 


24 


27 


31 


•18 


15136 


15171 


15205 


15241 


15276 ^ 


^ 7 


II 


14 


18 


21 


25 


28 


32 


•19 


15488 


15524 


15560 


15596 


15631 A 


^ 7 


11 


14 


18 


22 


25 


29 


32 


•20 


15849 


15885 


15922 


15959 


15996 A 


^ 7 


II 


15 


18 


22 


26 


29 


33 


•21 


16218 


16255 


16293 


16331 


16368 A 


8 


1 1 


15 


19 


23 


26 


30 


34 


•22 


16596 


16634 


16672 


16711 


16749 A 


^ 8 


II 


15 


19 


23 


27 


31 


34 


•23 


16982 


17022 


1 706 1 


1 7 100 


17140 A 


^ 8 


12 


16 


20 


24 


28 


32 


35 


•24 


17378 


17418 


17458 


17498 


17539 A 


8 


12 


16 


20 


24 


28 


32 


36 


•25 


17783 


17824 


17865 


17906 


I'jgA'j 4 


[ 8 


12 


16 


21 


25 


29 


33 


37 


•26 


18197 


18239 


18281 


18323 


18365 4 


8 


13 


17 


21 


25 


30 


34 


38 


•27 


1 862 1 


18664 


18707 


18750 


18793 A 


9 


13 


17 


22 


26 


30 


34 


39 


•28 


19055 


19099 


19143 


19187 


19231 A 


9 


13 


18 


22 


26 


31 


35 


40 


•29 


19498 


19543 


19588 


19634 


19679 5 


9 


14 


18 


23 


27 


32 


36 


41 


•30 


19953 


19999 


2004s 


20091 


20137 5 


9 


14 


18 


23 


28 


32 


37 


42 


•31 


20417 


20464 


20512 


20559 


20606 5 


9 


14 


19 


24 


28 


33 


3» 


43 


•32 


20893 


20941 


20989 


21038 


21086 5 


10 


15 


19 


24 


29 


34 


39 


44 


•33 


21380 


21429 


21478 


21528 


21577 5 


10 


15 


20 


25 


30 


35 


40 


44 


•34 


21878 


21928 


21979 


22029 


22080 5 


10 


15 


20 


2S 


30 


35 


40 


46 


•35 


22387 


22439 


22491 


22542 


22594 5 


10 


16 


21 


26 


31 


36 


41 


47 


•36 


22909 


22961 


23014 


23067 


23121 5 


1 1 


16 


21 


27 


3^ 


37 


42 


48 


•37 


23442 


23496 


23550 


23605 


23659 5 


11 


16 


22 


27 


33 


3» 


44 


49 


•38 


23988 


24044 


24099 


24155 


24210 6 


1 1 


17 


7 T 


28 


33 


39 


44 


50 


•39 


24547 


24604 


24660 


24717 


24774 6 


1 1 


17 


23 


28 


34 


40 


45 


51 


•40 


25119 


25177 


2523s 


25293 


25351 6 


12 


17 


23 


29 


35 


41 


47 


52 


•41 


25704 


25763 


25823 


25882 


25942 6 


12 


18 


24 


30 


36 


42 


48 


54 


•42 


26303 


26363 


26424 


26485 


26546 6 


12 


18 


24 


30 


36 


43 


49 


55 


•43 


26915 


26977 


27040 


27102 


27164 6 


12 


19 


25 


31 


37 


44 


50 


56 


•44 


27542 


27606 


27669 


277Z1> 


27797 6 


13 


19 


26 


32 


38 


45 


51 


57 


•45 


28184 


28249 


28314 


28379 


2844s 7 


13 


20 


26 


33 


39 


46 


52 


59 


•46 


28840 


28907 


2^97i 


29040 


29107 7 


13 


20 


27 


35 


40 


47 


53 


60 


•47 


29512 


29580 


29648 


29717 


29785 7 


14 


21 


27 


34 


41 


48 


55 


62 


•48 


30200 


30269 


30339 


30409 


30479 7 


14:211 


28 


35 


42 


49 


56 


63 


•49_ 


30903 


30974 


31046 


31117 


31189 7 


14I21I 


29 


36 


43 


50 


57 


64 



ANTILOGARITHMS 





5 


6 


7 


8 


9 


I 

2 


2 
5 


3 
"7 


4 
9 


5 
12 


6 
14 


7 
16 


8 
19 


9 


•00 


10116 


IOI39 


10162 


10186 


10209 


21 


•01 


10351 


10375 


10399 


10423 


10447 


2 


5 


7 


10 


12 


14 


17 


19 


22 


•02 


10593 


I06I7 


10641 


10666 


1 069 1 


2 


5 


7 


10 


12 


15 


17 


20 


22 


•03 


10839 


10864 


10889 


10914 


10940 


3 


5 


8 


10 


13 


15 


18 


20 


23 


•04 


1 1092 


I II 17 


11143 


11169 


11194 


3 


5 


8 


10 


13 


15 


18 


20 


n 


•OS 


1 1350 


1 1376 


1 1402 


1 1429 


1 1455 


3 


5 


8 


II 


13 


16 


18 


21 


24 


•o6 


11614 


II64I 


11668 


11695 


11722 


3 


5 


8 


1 1 


14 


16 


19 


22 


24 


•07 


11885 


II9I2 


11940 


11967 


11995 


3 


6 


8 


11 


14 


17 


19 


22 


25 


•o8 


12162 


12190 


12218 


12246 


12274 


3 


6 


8 


11 


14 


17 


20 


23 


25 


•09 


12445 


12474 


12503 


12531 


12560 


3 


6 


9 


12 


14 


17 


20 


23 


26 


•10 


12735 


12764 


12794 


12823 


12853 


3 


6 


9 


12 


15 


18 


21 


24 


26 


•11 


13032 


13062 


13092 


13122 


13152 


3 


6 


9 


12 


15 


18 


21 


24 


^7 


•12 


13335. 


13366 


13397 


13428 


13459 


3 


6 


9 


12 


16 


19 


22 


25 


28 


•13 


13646 


^1^71 


13709 


13740 


13772 


3 


6 


9 


13 


16 


19 


22 


25 


28 


•14 


13964 


13996 


14028 


14060 


14093 


3 


6 


10 


13 


16 


19 


^l 


26 


29 


•15 


14289 


14322 


14355 


14388 


14421 


3 


7 


10 


13 


17 


20 


23 


26 


30 


•i6 


14622 


14655 


14689 


14723 


14757 


3 


7 


10 


14 


17 


20 


24 


27 


30 


•17 


14962 


14997 


15031 


15066 


15101 


3 


7 


10 


14 


17 


21 


24 


28 


31 


•i8 


15311 


15346 


15382 


15417 


15453 


4 


7 


11 


14 


18 


21 


25 


28 


32 


•19 


15668 


15704 


15740 


15776 


15812 


4 


7 


11 


14 


18 


22 


25 


29 


11 


•20 


16032 


16069 


16106 


16144 


16181 


4 


7 


II 


15 


19 


22 


26 


30 


33 


•21 


16406 


16444 


16482 


16520 


16558 


4 


8 


11 


15 


19 


23 


27 


30 


34 


•22 


16788 


16827 


16866 


16904 


16943 


4 


8 


12 


16 


19 


'-I 


27 


31 


35 


•23 


17179 


17219 


17258 


17298 


17338 


4 


8 


12 


16 


20 


24 


28 


32 


36 


•24 


17579 


17620 


17660 


17701 


17742 


4 


8 


12 


16 


20 


24 


29 


33 


17 


•25 


17989 


18030 


18072 


18113 


18155 


4 


8 


12 


17 


21 


25 


29 


33 


37 


•26 


18408 


18450 


18493 


18535 


18578 


4 


9 


13 


17 


21 


26 


30 


34 


38 


•27 


18836 


18880 


18923 


18967 


19011 


4 


9 


13 


18 


22 


26 


31 


35 


39 


•28 


19275 


19320 


19364 


19409 


19454 


4 


9 


13 


18 


22 


27 


31 


36 


40 


•29 


19724 


19770 


19815 


19861 


19907 


5 


9 


14 


18 


23 


27 


32 


Z7 


41 


•30 


20184 


20230 


20277 


20324 


20370 


5 


9 


14 


19 


23 


28 


33 


37 


42 


•31 


20654 


20701 


20749 


20797 


20845 


5 


10 


14 


19 


24 


29 


IZ 


38 


43 


•32 


21135 


21184 


21232 


21281 


21330 


5 


10 


15 


20 


25 


29 


34 


39 


44 


•33 


21627 


21677 


21727 


^-^m 


21827 


5 


10 


15 


20 


25 


30 


35 


40 


45 


•34 


22131 


22182 


22233 


22284 


22336 


5 


10 


15 


20 


26 


31 


36 


41 


46 


•35 


22646 


22699 


22751 


22803 


22856 


5 


II 


16 


21 


26 


32 


37 


42 


47 


•36 


23174 


23227 


23281 


^ms 


23388 


5 


11 


16 


21 


27 


32 


38 


43 


48 


•37 


23714 


23768 


2ZZ21 


23878 


23933 


5 


II 


16 


22 


27 


^^ 


38 


44 


49 


•38 


24266 


24322 


24378 


24434 


24491 


6 


1 1 


17 


22 


28 


34 


39 


45 


51 


•39 


24831 


24889 


24946 


25003 


25061 


6 


12 


17 


23 


28 


35 


40 


46 


52 


•40 


2S4IO 


25468 


25527 


25586 


25645 


6 


12 


18 


24 


29 


35 


41 


47 


53 


•41 


26002 


26062 


26122 


26182 


26242 


6 


12 


18 


24 


30 


36 


42 


48 


54 


•42 


26607 


26669 


26730 


26792 


26853 


6 


12 


18 


25 


31 


17 


43 


49 


55 


•43 


27227 


27290 


27353 


27416 


27479 


6 


13 


19 


25 


32 


38 


44 


50 


57 


•44 


27861 


27925 


27990 


28054 


28119 


6 


13 


19 


26 


32 


39 


45 


52 


S8 


•45 


28510 


28576 


28642 


28708 


28774 


7 


13 


20 


26 


33 


40 


46 


53 


59 


•46 


29174 


29242 


29309 


29376. 


29444 


7 


14 


20 


^7 


34 


41 


47 


54 


61 


•47 


29854 


29923 


29992 


30061 


30130 


7 


14 


21 


28 


35 


42 


48 


55 


62 


•48 


30549 


30620 


30690 


30761 


30832 


7 


14 


21 


28 


35 


42 


50 


57 


64 


•49 


31261 


31333 


314OS 


31477 


31550 


7 


14 


21 


29 


36 


43 


51 


58 


65 



Mean Differences 



II 



Antilogarithms 



Mean Differences 








1 


2 


3 


4 


I 2 


3 


4 


5 


6 


7 


8 


9 


•SO 


31623 


3T696 


31769 


31842 


31915 


~7 


IS 


^ 


29 


37 


44 


SI 


59 


66 


•SI 


32359 


32434 


32509 


32584 


32659 


8 


15 


^3 


30 


3S 


45 


53 


60 


68 


•S2 


33113 


33189 


33266 


33343 


33420 


8 


15 


23 


31 


38 


46 


53 


61 


69 


'S3 


33884 


33963 


34041 


341 19 


34198 


8 


16 


24 


31 


39 


47 


55 


63 


71 


•54 


34674 


34754 


34834 


34914 


34995 


8 


16 


24 


32 


40 


48 


56 


64 


72 


•ss 


35481 


35563 


35645 


35727 


35810 


8 


16 


25 


53 


41 


49 


S8 


66 


74 


•S6 


36308 


36392 


36475 


36559 


36644 


8 


17 


2q 


34 


42 


50 


59 


67 


76 


•S7 


37154 


37239 


37325 


37411 


37497 


9 


17 


26 


34 


43 


52 


60 


69 


78 


•S8 


38019 


38107 


38194 


38282 


38371 


9 


18 


26 


35 


44 


53 


62 


70 


79 


•S9 


38905 


38994 


39084 


39174 


39264 


9 


18 


27 


36 


45 


54 


63 


72 


81 


•60 


3981 1 


39902 


39994 


40087 


40179 


9 


18 


28 


37 


46 


55 


65 


74 


83 


•61 


40738 


40832 


40926 


41020 


41115 


9 


19 


28 


38 


47 


57 


66 


76 


85 


•62 


41687 


41783 


41879 


41976 


42073 


10 


19 


29 


39 


48 


58 


67 


77 


87 


•63 


42658 


42756 


42855 


42954 


43053 


10 


20 


30 


40 


49 


59 


69 


79 


89 


•&4 


43652 


43752 


43853 


43954 


44055 


10 


20 


30 


40 


51 


61 


71 


81 


91 


•6s 


44668 


44771 


44875 


44978 


45082 


10 


21 


31 


41 


52 


62 


73 


83 


93 


•66 


45709 


45814 


45920 


46026 


46132 


1 1 


21 


32 


42 


53 


63 


74 


85 


95 


•67 


46774 


46881 


46989 


47098 


47206 


1 1 


22 


32 


43 


54 


65 


76 


86 


97 


•68 


47863 


47973 


48084 


48195 


48306 


1 1 


22 


33 


44 


55 


66 


78 


89 


100 


.69 


48978 


49091 


49204 


49317 


49431 


1 1 


23 


34 


45 


57 


68 


79 


91 


102 


•70 


SOI 19 


50234 


50350 


50466 


50582 


12 


23 


35 


46 


58 


70 


81 


93 


104 


•71 


51286 


51404 


51523 


51642 


51761 


12 


24 


36 


48 


59 


71 


83 


95 


107 


•72 


52481 


52602 


52723 


52845 


52966 


12 


24 


36 


49 


61 


73 


85 


97 


109 


•73 


53703 


53827 


53951 


54075 


54200 


12 


25 


37 


50 


62 


75 


87 


100 


1 12 


•74 


54954 


55081 


55208 


55335 


55463 


13 


25 


38 


51 


64 


76 


89 


102 


114 


•7S 


56234 


56364 


56494 


56624 


56754 


13 


26 


39 


52 


65 


78 


91 


104 


117 


.76 


57544 


57677 


57810 


57943 


58076 


13 


27 


40 


53 


67 


80 


93 


107 


120 


•77 


58884 


59020 


59156 


59293 


59429 


14 


27 


41 


55 


68 


82 


95 


109 


123 


•78 


60256 


60395 


60534 


60674 


60814 


14 


28 


42 


56 


70 


84 


98 


112 


126 


•79 


61660 


61802 


61944 


62087 


62230 


14 


29 


43 


57 


71 


86 


100 


114 


128 


•80 


63096 


63241 


63387 


63533 


63680 


15 


29 


44 


58 


73 


88 


102 


117 


131 


•81 


64565 


64714 


64863 


65013 


65163 


15 


30 


45 


60 


75 


90 


105 


120 


135 


•82 


66069 


66222 


<^^^374 


66527 


66681 


15 


31 


46 


61 


77 


92 


107 


122 


138 


•83 


67608 


67764 


67920 


68077 


68234 


16 


31 


47 


63 


78 


94 


no 


125 


141 


•84 


69183 


69343 


69502 


69663 


69823 


16 


32 


48 


64 


80 


96 


112 


128 


144 


•85 


70795 


70958 


71121 


71285 


71450 


16 


33 


49 


66 


82 


98 


IIS 


131 


147 


•86 


72444 


7261 1 


72778 


72946 


73114 


17 


34 


50 


(^7 


84 


101 


117 


134 


151 


•87 


74131 


74302 


74473 


74645 


74817 


17 


34 


Si 


69 


86 


103 


120 


137 


154 


•88 


75858 


76033 


76208 


76384 


76560 


18 


35 


S3 


70 


88 


105 


123 


140 


158 


•89 


77625 


77804 


77983 


78163 


7S343 


18 


36 


54 


7~ 


90 


108 


126 


144 


162 


•90 


79433 


79616 


79799 


79983 


80168 


18 


37 


55 


74 


92 


no 


129 


147 


166 


•91 


81283 


81470 


81658 


81846 


82035 


19 


38 


56 


75 


94 


113 


132 


151 


169 


•92 


83176 


83368 


83560 


83753 


83946 


19 


39 


58 


77 


96 


116 


135 


154 


174 


•93 


85114 


85310 


85507 


85704 


85901 


20 


39 


59 


79 


99 


118 


138 


158 


177 


•94 


87096 


87297 


87498 


87700 


87902 


20 


40 


61 


81 


101 


121 


141 


161 


182 


•95 


8912s 


89331 


89536 


89743 


89950 


21 


41 


62 


83 


103 


124 


144 


165 


186 


•96 


91201 


91411 


91622 


91833 


92045 


21 


42 


63 


84 


106 


127 


148 


169 


190 


•97 


93325 


93541 


93756 


93972 


94189 


22 


43 


65 


80 


108 


130 


151 


173 


195 


•98 


95499 


95719 


95940 


96161 


96383 


22 


44 


6() 


88 


1 1 1 


133 


155 


177 


199 


M 


97724 


97949 


98175 


98401 


98628 


23 


45 


68 


90 


113 


136 


158 


181 


204 



12 



ANTILOGARITHMS 





5 


6 


7 


8 


9 


^\l 


3 


4 


5 


6 


7 


\' 


9 


•so 


31989 


32063 


32137 


32211 


32285 


7 


15 


22 


30 


37 


44 


52 


59 


67 


•51 


32734 


32810 


32885 


32961 


33037 


8 


15 


23 


30 


38 


45 


53 


61 


68 


•52 


33497 


33574 


33651 


33729 


33806 


8 


15 


23 


31 


39 


46 


54 


62 


70 


'53 


34277 


34356 


34435 


34514 


34594 


8 


16 


24 


32 


40 


48 


55 


C^3 


71 


•54 


35075 


35156 


35237 


35318 


35400 


8 


16 


24 


32 


41 


49 


57 


65 


73 


•55 


35892 


35975 


36058 


36141 


36224 


8 


17 


25 


33 


42 


50 


S8 


67 


75 


•S6 


36728 


36813 


36898 


36983 


37068 


9 


17 


26 


34 


43 


51 


60 


68 


77 


•57 


37584 


17670 


Z77S7 


37844 


37931 


9 


17 


26 


35 


44 


52 


61 


70 


78 


•58 


38459 


38548 


38637 


38726 


38815 


9 


18 


27 


36 


45 


54 


62 


71 


80 


•59 


39355 


39446 


39537 


39628 


39719 


9 


18 


27 


36 


46 


55 


64 


73 


Sjj 


•6o 


40272 


40365 


40458 


40551 


40644 


9 


19 


28 


37 


47 


56 


6S 


75 


84 


•6i 


41210 


41305 


41400 


41495 


41591 


10 


19 


29 


38 


48 


57 


67 


76 


86 


•62 


42170 


42267 


42364 


42462 


42560 


10 


20 


29 


39 


49 


59 


68 


7S 


88 


•63 


43152 


43251 


43351 


43451 


43551 


10 


20 


30 


40 


50 


60 


70 


80 


90 


•64 


44157 


44259 


44361 


44463 


44566 


10 


20 


31 


41 


51 


61 


72 


82 


92 


•65 


45186 


45290 


45394 


45499 


45604 


10 


21 


31 


42 


52 


63 


73 


84 


94 


•66 


46238 


46345 


46452 


46559 


46666 


II 


21 


32 


43 


54 


64 


75 


86 


96 


•67 


47315 


47424 


47534 


47643 


47753 


II 


22 


33 


44 


55 


66 


77 


88 


99 


•68 


48417 


48529 


48641 


48753 


48865 


II 


22 


34 


45 


56 


(^7 


79 


90 


lOI 


.69 


49545 


49659 


49774 


49888 


50003 


II 


23 


34 


46 


57 


69 


80 


92 


103 


•70 


50699 


50816 


50933 


51051 


51 168 


12 


23 


35 


47 


59 


70 


82 


94 


106 


.71 


51880 


52000 


52119 


52240 


52360 


12 


24 


36 


48 


60 


72 


84 


96 


108 


•72 


53088 


53211 


^ZZIZ 


53456 


53580 


12 


25 


37 


49 


62 


74 


86 


98 


III 


•73 


54325 


54450 


54576 


54702 


54828 


13 


25 


38 


50 


63 


75 


88 


100 


113 


•74 


55590 


55719 


55847 


55976 


56105 


13 


26 


39 


52 


64 


77 


90 


103 


116 


•75 


56885 


57016 


57148 


57280 


57412 


13 


26 


40 


S3 


66 


79 


92 


105 


119 


.76 


58210 


58345 


58479 


58614 


58749 


13 


27 


40 


54 


67 


81 


94 


108 


121 


•77 


59566 


59704 


59841 


59979 


601 17 


14 


28 


41 


55 


69 


Si 


97 


no 


124 


•78 


60954 


61094 


61235 


6\Z7^ 


61518 


14 


28 


42 


56 


71 


85 


99 


113 


127 


•79 


^^Z7l 


62517 


62661 


62806 


62951 


14 


29 


43 


58 


72 


87 


lOI 


116 


130 


.80 


63826 


63973 


641 2 1 


64269 


64417 


15 


30 


44 


59 


74 


88 


104 


118 


133 


•81 


^sm 


65464 


65615 


65766 


65917 


15 


30 


45 


60 


76 


91 


106 


121 


136 


•82 


66834 


66988 


67143 


67298 


67453 


15 


31 


46 


62 


77 


93 


108 


124 


139 


•83 


68391 


68549 


68707 


68865 


69024 


16 


32 


48 


63 


79 


95 


III 


127 


143 


•84 


69984 


70146 


70307 


70469 


70632 


16 


32 


49 


65 


81 


97 


114 


130 


146 


•85 


71614 


71779 


71945 


721 1 1 


72277 


17 


33 


50 


66 


83 


100 


116 


133 


149 


•86 


73282 


73451 


73621 


73790 


73961 


17 


34 


51 


6S 


84 


102 


119 


136 


153 


•87 


74989 


75162 


75336 


75509 


75683 


17 


35 


52 


70 


87 


104 


122 


139 


156 


•88 


76736 


76913 


77090 


77268 


77446 


18 


36 


53 


71 


89 


107 


124 


142 


160 


.89 


78524 


78705 


78886 


79068 


79250 


18 


36 


55 


73 


91 


109 


127 


145 


164 


•90 


80353 


80538 


80724 


80910 


81096 


19 


37 


56 


74 


93 


112 


130 


149 


167 


•91 


82224 


82414 


82604 


82794 


82985 


19 


3S 


57 


76 


95 


114 


133 


1^2 


171 


.92 


84140 


Hill 


84528 


84723 


84918 


19 


39 


58 


78 


97 


117 


136 


i=;6 


175 


•93 


86099 


86298 


86497 


86696 


86896 


20 


40 


60 


80 


100 


120 


140 


160 


179 


•94 


88105 


88308 


88512 


88716 


88920 


20 


41 


61 


82 


102 


122 


143 


163 


184 


•95 


90157 


90365 


90573 


90782 


90991 


21 


42 


63 


84 


104 


125 


146 


167 


188 


.96 


922^ M70 

-^ -24 


92683 


92897 


93111 


21 


43 


64 


85 


107 


128 


150 


171 


192 


.n-it 


94842 


95060 


95280 


22 


44 


66 


87 


109 


131 


153 


175 


197 


W > ^ ' 


97051 


97275 


97499 


22 


45 


67 


90 


112 


134 


157 


179 


201 


^- 


_.j993i2| 


99541 


99770 


13_ 


46 


69 


9^ 


115 


137 


160 


183_ 


206 
















M 


ean ] 


Diffe 


renc 


2S 





0° NATURAL FUNCTIONS 

Differences are given for every lo'. Intermediate values can be found by n 
method of proportional parts ; e. g. : — 

To find tan 43° 56' and cos 37° 34' 

tan 43° 5o'= -96008 cos 37° 3°'= 79335 1 

+ diff. for 6'= 337 -diff. for4'=- 71 f 

.-. tan 43° 5^'= -96345 ••• cos 37° 34'= -79264 

When there is no entry in the difference column, the value of the function char 
too rapidly for correct interpolation by proportional parts. C4reater accuracy is t 
obtained by expressing the function in terms of the sine and cosine. 

To find tan 67° 23' 

tan 67° 20'= 2-39449 Diff. for 10' = 1972 

by proportional parts, diff. for 3'= 592 

This gives tan 67° 23'= 2-40041. (The correct value is 2-40038.) 



Subtract differences when dealing with co-functions 



0° 

10' 

20' 

30' 

40' 

';o^ 

i'' 

10' 

20' 

30' 

40' 

50' 

2° 

10' 

20' 

30' 

40' 

so' 

3° 



•00000 

•00291 
•00582 
•00873 
•o I 1 64 
•01454 

•01745 

•02036 
•02327 
•02618 
•02908 
•03199 

•0349Q 

•03781 
•0407 1 
•04362 
•04653 
■04943 
05234 

cosine 



291 

291 
291 
291 
290 
291 
291 
291 
291 
290 
291 
291 
291 
290 
291 
291 
290 
291 

D 



00 

34378 
171-89 

114-59 
85-946 

68-757 
57-299 

49-114 
42-976 
38-202 
34-382 
'31-258 

28-654 

26-451 
24-562 
22-926 
2 1 -494 
20-230 
19-107 
secant 



tangent 
-00000 

•00291 
•00582 
•00873 
•01 164 
•01455 
•01746 
•02037 
•02328 
•02619 
•02910 
■03201 
03492 
■03783 
■04075 
•04366 
■04658 
04949 
05241 
D cotangent 



291 

291 
291 
291 
291 
291 
291 
291 
291 
291 
291 
291 
291 
292 
291 
292 
291 
292 

^^ 



cotangent 



00 

343-77 
171-89 

114-59 
85-940 
68-750 
57290 
49-104 
42-964 
38^188 
34-368 
31-242 
28636 
26-432 
24-542 
2 2 -904 
21-470 

20-206 
I9081 
tang(;nt 



secant 



I -00000 

I -ooooo 

I -00002 
I -00004 
I -00007 
I -000 1 I 

I 000 I 5 

I ^0002 1 
1^00027 
I -00034 
I -00042 
I -0005 I 
I 0006 1 
I -00072 
1-00083 
I -00095 
I -00108 

I -OO I 2 2 
I -00137 
cosecant 



000 

002 
002 
003 
004 
004 

005 

006 
007 
008 
009 
010 
Oil 

01 1 
012 

013 

014 

015 



I -ooooo 

•99998 
-99996 
-99993 
-99989 
-99985 
•99979 
•99973 

-99966 

•99958 

-99949 
-99939 

•99929 
•99917 
•99905 
-99892 
-99878 
-99863 



000 

002 
002 
003 
004 
004 
006 
006 
007 
008 
009 
010 

GIG 

012 
012 
013 
014 
015 



|9C 

5< 
4c 

3C 

2C 
IC 

8S 
5t 
4C 
3c 

2C 
IC 

50 

4oi 
30 
20 
10 
87 



0^ LOGARITHMIC FUNCTIONS 

The values given here are the true logarithms ; the characteristic is not 
increased by lo as in many tables. 

Differences are given for every lo'. Intermediate values can be found by 
the method of proportional parts. 

The differences for the logarithm of a function and of the reciprocal of the 
function are the same in magnitude but opposite in sign. 

When there is no entry in the difference column, the rate of change of the 
logarithm changes too rapidly for correct interpolation by proportional parts. 

The following rules may be used when the angle is small : — 

Log sine. Add &6S$S7 to the log of the angle expressed in seconds and 

subtract ^ of^tlie log secant. 
Log tan. Add 6-68557 to the log of the angle expressed in seconds and 

add ^ of the log secant. 
When the log sine is given, the angle is found in seconds by adding 

5-31443 to the log sine and ^ of the corresponding log secant (found 

in the ordinary way). 
When the log tan is given, the angle is found in seconds by adding 

5*31443 to the log tan and subtracting | of the corresponding log 

secant (found in the ordinary way). 





Subtract differences when dealing with co 


-functions 






log sin 


D 


log cosec 


log tan 


D |log cotan 


log sec 


D ] log COS 




d^ 


— 00 




00 


— 00 




00 


0-00000 




0-00000 


90° 


10' 


3-46373 




2-53627 


3-46373 




2-53627 


0-00000 


001 


o-ooooo 


50' 


20' 


376475 




2-23525 


3-76476; 


2-23524 


0-0000 1 


001 


1-99999 


40' 


30' 


3-94084 




2-05916 


3-94086 




2-05914 


-00002 


^; 11-99998 


30' 


40' 


2-06578 




1-93422 


2-06581 


jl-93419 


0-00003 


002 


1-99997 


20' 


50' 


2-16268 




1-83732 


2-16273 


1-83727 


0-00005 


002 


1-99995 


10' 


r 


2-24186 


1-75814 


2-24192 


1-75808 


0-00007 


002 


1-99993 


89"^ 


10' 


2-30879 1 


I -69 1 2 1 


2-30888 


i I -691 12 


0-00009 


00 i I -9999 1 


50' 


20' 


2-366781 


1-63322 


2-36689 


I-633II 


0-00012 


003 

00 ■? 


1-99988 


40' 


30' 


2-41792, 


1-58208 


2-41807 


1-58193 


0-00015 


1-99985 


30' 


40' 


2-463661 


1-53634 


2-46385 




1-53615 


0-00018 


^l ■•9»«-'| 


20' 


50' 


2-50504 


I -49496 


2-50527 




1-49473 


0-00022 


004 
004 


1-99978 


10' 


2^ 


2-S4282I 


1-45718 


2-54308 




1-45692 


000026 


1-99974 


88^ 


10' 


2-57757 


1-42243 


2-57788 




1-42212 


0-00031 


1-99969 


50' 


20' 


2-60973 


1-39027 


2-61009 




I -38991 


0-00036 


005 ^-99964 


40' 


30' 


2-63968. 


1-36032 


2-64009 




1-35991 


0-00041 


o^!i-99959 


30' 


40' 


2-66769! 


1-33231 


2-66816 




1-33184 


o-o(J047 


0^^99953 


20' 


50' 


2-69400' 


1-30600 


2-69453 


j I -30547 


o-(;<KJ53 


006 ^-99947 


10' 


3" 


2-71880' 


1-28120 


2-71940 


11-28060 


000060 


i -99940 


ST 




log COS j D 


log sec 


log cotan 


D 1 log tan 


log cosec 


D 


log sin 





87= 



3° NATURAL FUNCTIONS 



05234 

05524 
05814 
06105 
06395 
06685 
06976 
07266 
07556 
07846 
08136 
08426 
08716 
09005 
09295 

09585 

09874 

0164 

0453 

0742 
1031 
1320 
1609 
1898 
2187 
2476 
2764 
3052 
3341 
3629 

3917 

4205 
4493 
4781 
5069 
5356 
5643 

5931 
6218 

6505 
6792 
7078 
736s 



290 

290 
291 
290 
290 
291 
290 
290 
290 
290 
290 
290 
289 
290 
290 
289 
290 
289 
289 
289 
289 
289 
289 
289 
289 
288 
288 
289 
288 
288 
288 
288 
288 
288 
287 
287 
288 
287 
287 
287 
286 
287 



D 



19-1073 

18-1026 
17-1984 
16-3804 
15-6368 
14-9579 
14-3356 
13-7631 
13-2347 
127455 
12-2913 
11-8684 

11-4737 

11-1046 
10-7585 
IO-4334 
10-1275 
9-83912 
956677 
9-30917 
9-06515 

8-83367 
8-61379 
8-40466 

8-20SSI 

8-01565 

7-83443 
7-66130 

7-49571 

7-18530 

7-03962 
6-89979 
6-76547 
6-63633 
6-51208 

639245 

6-27719 
6-16607 
6-05886 
5-95536 
5-85539 
5 75877 



tangent 



D 



-05241 

•05533 
-05824 
•061 16 
•06408 
•06700 
-06993 
•07285 
•07578 
-07870 
-08163 
•08456 

-08749 

•09042 

-09335 
•09629 
•09923 
-IO216 
-IO51O 
•10805 
•I 1099 

-II394 
-I1688 
-II983 
-12278 

-12574 
•12869 
•13165 
•I 3461 
•13758 
•14054 

-1435 I 
• 1 4648 

-14945 
•15243 
•15540 

•15838 

•I 61 37 
•16435 
•16734 
•17033 
'^7Zr:s 
•17633 



292 

291 
292 
292 
292 
293 
292 

293 
292 

293 
293 
293 
293 

293 
294 
294 

293 
294 

295 

294 
295 
294 
295 
295 
296 

295 
296 
296 

297 
296 

297 

297 
297 
298 
297 
298 
299 
298 
299 
299 
300 
300 



cotangent] D 



cotangent 



I9081I 

18^0750 
17-1693 
16-3499 
I 5 -6048 
14-9244 
14-3007 
13-7267 

13-1969 
12-7062 
12-2505 
11-8262 
Iit430i 
11-0594 
10-7119 
10-3854 
10-0780 
978817 
9-S1436 

9-25530 
9-00983 
8-77689 

8-55554 
8 -34496 

8-14435 

7-95302 
7-77035 
7-59575 
7-42871 
7-26873 
7^ii537 
6-96823 
6-82694 
6-691 16 
6-56055 
6^43484 
631375 
6-19703 
6-08444 
5-97576 
5-87080 
5-76937 
567128 



tangent 



I 00137 

I -00153 
I -00 1 69 
I -001 87 
1-00205 
1-00224 
1-00244 
1-00265 
1-00287 
1-00309 
I -00333 
1-00357 
I 00382 
I -00408 
I -00435 
I -00463 
I -0049 1 
■I -005 2 I 

I -00551 

1-00582 
I -006 1 4 
I -00647 
I -0068 I 
I -007 I 5 
I 0075 1 
1-00788 
1-00825 
1-00863 
I -00902 
I -00942 
I 00983 
1-01024 
I -01067 
I -O I I I I 
I-OII55 
I -01 200 
I 01247 
1-01294 
I -01 342 
I -01 39 1 
I -01 440 
I -01 49 1 

I 01543 



D 



016 

016 
018 
018 
019 
020 
021 
022 
022 
024 
024 
025 
^f026 

027 
028 
028 
030 
030 
031 
032 
033 
034 
034 
036 

037 

037 
038 
039 

040 
041 
041 

043 
044 
044 
045 
047 
047 
048 

049 
049 

051 
052 



99863 

99847 
99831 

99813 

99795 
99776 
99756 

99736 
■99714 
■99692 
■99668 
■99644 
99619 

■99594 
■99567 
•99540 

■995 1 1 
•99482 

•99452 

-99421 
-99390 
-99357 
•99324 
-99290 
•99255 
•99219 
•99182 
-99144 
-99106 
•99067 
•99027 
•98986 

•98944 
•98902 
■98858 
■98814 
98769 
■98723 
■98676 
•98629 
■98580 
98531 
98481 



80^ 



LOGARITHMIC FUNCTIONS 





loo^ sin 


D 


log cosec 


log tan 




log cotan 
1-28060 


lo^ sec 


D 


log cos 




3" 


271880 




1-28120 


271940 




0-00060 


006 


i -99940 


87^ 


lO' 


2-74^26 




1-25774 


2-74292 




1-25708 


■ J -00066 


008 


1-99934 


50' 


20' 


276541 




1-23549 


2-76525 




1-2 347 S 


0-00074 


007 
Of 18 


1-99926 


40' 


30' 


278568 




1-21432 


2-78649 




1-21351 


0-0008 1 


1-99919 


30' 


4''' 


2-80585 




1-19415 


2-80674 




1-19326 


0-00089 


LHJO 
008 


T -999 11 


20' 


;o' 


2-S2SI3 




1-17487 


2-82610 




1-17390 


0-00097 


009 
009 
009 
010 


1-99903 


10' 


4^ 

10' 


284358 

2-86128 




1-15642 

1-13872 


2 84464 

5-60243 




1-15536 

1-13757 


000106 

0-001 15 


199894 

1-99S85 


86^ 

50' 


20' 


2-87829 




1-12171 


2-87953 




I -12047 


0-00124 


1-99876 


40' 


3<J' 


2-89464 




1-10536 


2-89598 




1-10402 


0-00134 


T r> 


1-99866 


30; 


40' 


2-91040 




I -08960 


2-91185 




1-08815 


0-00144 


(J lU 

Oil 


1-99856 


20' 


Sf'j' 


5-92561 




1-07439 


2-92716 




1-07284 


0-00155 


Oil 


1-99S45 


10' 


5" 


2 94030 




105970 


2-94195 




1-05805 


000166 


Oil 


1-99834 


85^ 


10' 


^•95450 




1-04550 


2-95627 




1-04373 


0-00177 


r> T T 


1-99823 


50' 


20' 


2-96825 




1-03175 


2-97013 




1-02987 


0-00188 


L) 1 1 
012 
013 
012 


1-99812 


40' 


30' 


2-98157 




1-01843 


2-98358 




1-01642 


0-00200 


1-99800 


30' 


40' 


2-99450 




1-00550 


2-99662 




1-00338 


0-00213 


1-99787 


20' 


;o' 


I -00704 




0-99296 


1-00930 




0-99070 


0-00225 


014 
013 


1-99775 


10' 


6^~ 


101923 




098077 


i-02162 




097838 


000239 


i -99761 


84° 


10' 


T -03 109 




0-96891 


1-03361 




0-96639 


0-00252 


1-99748 


50' 


20' 


1-04262 




0-95738 


1-04528 




0-95472 


0-00266 


014 
014 

015 
r\T - 


1-99734 


40' 


30' 


1-05386 




0-94614 


1-05666 




0-94334 


0-00280 


1-99720 


30' 


40' 


I -0648 1 




0-93519 


1-06775 




0-93225 


0-00295 


1-99705 


20' 


50' 


1-07548 




0-92452 


1-07858 




0-92142 


0-00310 


oio 

T ; 


1 -99690 


10' 


7=^ 


1-08589 




0-91411 


108914 




0-91086 


0-00325 


Ul3 
016 


199675 


83^ 


10' 


I -09606 




0-90394 


I -09947 




0-90053 


0-00341 


016 
016 


1-99659 


50' 


20' 


1-10599 


993 


0-89401 


I -10956 


987 
966 

945 
926 

908 

889 

873 
856 
840 
825 
811 


0-89044 


0-00357 


1-99643 


40' 


30' 


1-11570 


971 


0-88430 


I-II943 


o'88o57 


0-00373 


1-99627 


30' 


40' 

8= 

10' 


1-12519 
1-13447 
I 14356 

1-15245 


949 
928 
909 
889 
871 
854 

837 
821 
805 
790 


0-87481 
0-86553 
085644 

0-84755 


I -12909 
1-13854 
i- 14780 

1-15688 


0-87091 
0-86146 
085220 

0-84312 


0-00390 
0-00407 
000425 

0-00443 


017 
017 

018 

018 

018 
019 
019 


1-99610 
1-99593 
I -99575 
1-99557 


20' 
10' 
82° 

50' 


20' 


i-i6ii6 


0-83884 


T-16577 


0:83423 


0-00461 


1-99539 


40' 


3^' 


1-16970 


0-83030 


1-17450 


0-82550 


0-00480 


1-99520 


30' 


40' 


1-17807 


0-82193 


1-18306 


0-81694 


-00499 


1-99501 


20' 


50' 
9~ 


1-18628 
1-19433 


0-81372 
080567 


1-19146 
1-19971 


0-80854 
80029 


0-00518 
0-00538 


019 
020 

020 


1-99482 
1-99462 


10' 
81° 


10' 
20' 


1-20223 
1-20999 


776 
762 
748 


0-79777 
0-79001 


1-20782 
I -2 1 578 


796 

783 
769 


0-79218 
0-78422 


0-00558 
0-00579 


.021 
021 
021 
022 
022 

D 


I -99442 
1-99421 


50' 
40' 


30' 
40' 


1-21761 
1-22509 


078239 
077491 


1-22361 
I -23 1 30 


0-77639 
0-76870 


0-00600 
0-00621 


1 -99400 
1-99379 


30' 
20' 


^0' 
10"" 


1-23244 
1-23967 


735 
723 


0-76756 
0-76033 


1-23887 
1-24632 

log cotan 


757 
745 


0-76113 
0-75368 


0-00643 
00665 

log cosec 


1-99357 
1-99335 

log sin 


10' 
80 




log COS 


D 


log sec- 


D 


log tan 





80= 



17 



10'' NATURAL FUNCTIONS 



M 



17365 

■ I 765 1 

■17937 
•18224 
•18509 
■18795 
19081 
■19366 
•19652 

•19937 
•20222 
•20507 
•20791 
•21076 
•21360 
•21644 
•21928 
•22212 
•22495 
•22778 
•23062 

•23345 
•23627 
•23910 
•24192 

•24474 
•24756 
•2503S 
•25320 
•25601 
•25882 
•26163 
•26443 
•26724 
•27004 
•27284 

•27564 

•27843 
•28123 
•28402 
•28680 
•28959 

•29237 



286 

286 
287 
285 
286 
286 
285 
286 
285 
285 
285 
284 

285 

284 
284 
284 
284 
283 
283 
284 
283 
282 
283 
282 
282 
282 
282 
282 
281 
281 
281 
280 
281 
380 
280 
280 
279 
280 

279 

278 
279 
278 



D 



5-75877 

5-66533 

5-57493 
5-48740 
5-40263 
5-32049 
524084 

5-16359 
5-08863 

5-01585 
4-94517 
4-87649 
480973 

4-74482 
4^68167 
4-62023 
4^56o4i 
4^50216 

4-44541 

4-39012 
4-33622 
4^28366 

4-23239 
4-18238 

4-13357 

4-08591 
4-03938 
3-99393 
3^94952 
3^90613 

386370 

3-82223 
3^78166 
3-74198 

3-70315 
3-66515 

3 62796 

3-59154 
3^55587 
3-52094 
3-48671 
3-45317 
342030 



tangent 



17633 

17933 
18233 

18534 
18835 
19136 

19438 

19740 
20042 

20345 
20648 
20952 
21256 

21560 
21864 
22169 

22475 
22781 

23087 

23393 
23700 
24008 
24316 
24624 
24933 
25242 
25552 
25862 
26172 
26483 
26795 
27107 
27419 

28046 
28360 

28675 

28990 
29305 
29621 
29938 
30255 

30573 



otangont 



300 

300 
301 
301 
301 
302 
302 
302 
303 
303 
304 
304 
304 

304 
305 
306 
306 
306 
306 

307 
308 
308 
308 
309 
309 
310 
310 
310 

311 
312 

312 

312 
313 
314 
314 
315 
315 

315 
316 
317 
317 
318 



cotangent D 



567128 

5^57638 
5-48451 

5-39552 
5-30928 
5^22566 

5-14455 

5-06584 
4-98940 
4-91516 
4-84300 
4-77286 

4-70463 

4-63825 

4-57363 
4-51071 
4-44942 
4-38969 

433148 

4-27471 

4-21933 
4-16530 
4-11256 
4-06107 
401078 
3-96165 

3^91364 
3 •86671 
3-82083 
3-77595' 
373205 
3-68909 
3-64705 
3 60588 
3-56557 
3-52609 

3-48741 

3-44951 
3-41236 

3-37594 
3-34023 
3-30521 
327085 



tangent 



I-0I543 

i^oi595 
1-01649 
1-01703 
I •01758 
1-01815 
I 01872 
1-01930 
1-01989 
I -02049 
I -021 10 
I -02 1 7 1 
I 02234 
1-02298 
1-02362 
I ^02428 
I -02494 
1-02562 
I 02630 
1-02700 
1-02770 
1-02841 
1-02914 
1-02987 
I 03061 

1-03137 
I -032 1 3 
1-03290 
1-03368 
I -03447 
I 03528 
I ^03609 
1-03691 
I -03774 
1-03858 
I -03944 
I 04030 
1-04117 
I -04206 
1-04295 
1-04385 
I -04477 
I 04569 



052 

054 
054 
055 
057 
057 
058 

059 
060 
061 
061 
063 
064 
064 
066 
066 
068 
068 
070 
070 
071 
073 
073 
074 
076 
076 

077 
078 
079 
081 
081 
082 
083 
084 
086 
086 

087 

089 
089 
090 
092 
092 



I) 



98481 

98430 
98378 
98325 
98272 
98218 
98163 
98107 
98050 
97992 
^7934 
);875 
9781S 

97754 
97692 
97630 
97566 
97502 

97437 

97371 
97304 
97237 
97169 
97100 
97030 
96959 
96887 
96815 
96742 
96667 

96593 

965^7 
96440 
.96363 
96285 
96206 
96126 
96046 

95964 
95882 

95799 
95715 
95630 



73= 



18 



10= 



LOGARITHMIC FUNCTIONS 



log sin D 



71] 



23967 

•24677 
•25376 
•26063 
•26739 
•27405 
•28060 



I 

698 
687 
676 

666 
655 
64s 



635 
625 
616 

= 3/S 582 
-960 ^^^ 

■33534 ,66 

■5ATOO. ^ 



•29340 
•29966 
■305.8 
•31 

•3 



^-rlOO 
•3465 



•34100!^ g 



. -J - -7 - T 

537 



•35752 



•36289 'J' 
•37341 'i I 

4/:> 
■41300 ^g 

•^'^^' 464 
■-^''^'1458 

•43591 :|^3 
•44034 438 

■^^-^'"^ 433 

•44905 li' 

■45334/: 2^ 

-. . -_.o 424 



log coscc 

076033 

075323 

0-74624 

073937 
0-73261 
0-72595 
071940 

0-71295 

0-70660 
0-70034 
0-69418 
0^688 1 1 

0^682I2 

0^67622 
0-67040 
0-66466 
0-65900 

0-65342 
0-64791 

0^64248 
0-637II 
0-63181 
0-62659 
; 0-62 1 42 

061632 

0-61129 
0^60631 
0^60140 

0-59654 
0-59175 
0-58700 

0-58232 

;0-57768 
io-57310 
0-56857 
0-56409 
0-55966 
0-55528 
0-55095 
0-54666 
0-54242 
0-53822 
053406 



46594 

log COS D log sec 



log tan D log cotan 



•24632 

•26086 
•26797 
•27496 

•28186 

•28865 

■-9S3S 
•30195 
•30846 
•31489 
•32122 

•32747 

•33365 
•33974 
•34576 
•35170 
■3S7S7 
-36336 
•36909 
•37476 
•38035 
•38589 
•39136 

•39677 

•40212 
•40742 
•41266 
•41784 
•42297 
•42805 
•43308 
•43806 
•44299 
•44787 
•45271 

•4S7SO 

•46224 
•46694 
•47160 
•47622 
•48080 
48534 



733 

721 
711 
699 
690 
679 
670 
660 
651 
643 
633 
62 



075368 

0-74635 
0-73914 
0-73203 
0-72504 
0-71814 

0-71135 

0-70465 
0-69805 
0-69154 
0-68511 
0-67878 



579 
573 

567 
559 



6,8 067253 

^ ^ 0-6663 ^ 

5^9 0.66026 
601 , 

0-65424 

'f 0-64830 

0-64243 

o 63664 

0-63091 
0-62524 

^, 0-61965 

^^4 0-61411 

:)47 

541 

535 

530 

524 

518 

513 

508 

503 

498 

493 
488 

484 
479 
474 



0-60864 
060323 

0-59788 
0^59258 
0-58734 
io-58216 
'0-57703 

0-57195 

0^56692 
0^56194 
0-55701 
0-55213 
0-54729 

054250 

0-^3776 

^66 ^-53306 
^ 0^52840 
0-52378 
0-51920 
051466 



462 
458 
454 



log cotan 



log sec D log cos 



o 00665 

0^00687 
0^00710 
0-00733 
0-00757 
0-00781 
000805 
0-00830 
0-00855 
©•00881 
©•00907 

o -009 3 3 
o 00960 

©•00987 
0-01014 
©•01042 
©•01070 
©•01099 
001128 
©•01157 
0-01187 
©-01217 
0-01247 
©•01278 
001310 
©•©1341 
©•©1373 
©•01406 
©•01439 
©•©1472 
001506 
©•©154© 

<J-oi574 
©•©i6©9 
©•©1644 
©•0168© 
001716 
©•©1752 
©•01789 
©•01826 
©•01864 
©•©1902 
00 1940 

log cosic 



022 

023 
023 
024 
©24 
©24 
025 
025 
©26 
©26 
026 
027 
027 
027 
©28 
©28 
©29 
©29 
029 
03© 

©3© 
03© 
031 
032 

031 

032 
033 
033 
033 
034 
034 
034 
035 
035 
036 
©36 
036 
037 
037 
038 
038 
038 

u 



1-99335 

1-99313 
1-99290 
1-99267 
1^99243 
r992i9 

199195 

1-99170 
1-99145 
1-991 19 
1-99093 
I •99©67 
i 99040 

I -990 1 3 
1^98986 
T^98958 
1-98930 
i^989©i 
198872 

1-98843 
1-98813 
1-98783 
1-98753 
1^98722 

I 98690 

T^98659 
i^98627 

1-98594 
i^9856i 
i^98528 

I 98494 

I •9846© 
1-98426 
1-98391 
1-98356 
1-98320 
I 98284 
i^98248 
i^982i I 
1-98174 
I -98 1 36 
I -98098 
I 98060 



log 



50' 
40' 
30' 
20' 
10' 

79^ 

50^ 

40' 

30' 

20' 

I©' 

78^ 

50' 
40' 
30' 
20' 
10' 
77° 
50' 
40' 
30' 
20' 
I©' 
76' 

50' 
40' 
30' 
20' 
10' 
75' 
50' 
40' 
30' 
20' 
10' 
740 

50' 
40' 
30' 
20' 
10' 
73° 



73'' 



19 



17° NATURAL FUNCTIONS 



•29237 

•29515 
•29793 
•30071 

•30348 
•30625 

•30902 

•31178 
•31454 
•31730 
•32006 
•32282 

•32557 

•32832 
•33106 
•33381 
•33655 
33929 
•34202 

34475 
34748 
•35021 
•35293 
•35565 
•35837 
•36108 

■36379 
•36650 
•36921 
•37191 
37461 
37730 
37999 
38268 

38537 
38805 

39073 

39341 
39608 

39875 
40142 
40408 

40674 



278 

278 

278 

277 

277 

277 

276 

276 

276 

276 

276 

275 

275 

274 

275 

274 

274 

273 

273 

^71 

273 

272 

272 

272 

271 

271 

271 

271 

270 

270 

269 

269 

269 

269 

268 

268 

268 

267 

267 

267 

266 

266 



3 42030 

3-38808 
3-35649 
3-32551 
3-29512 
3-26531 
3-23607 

3-17920 

3-I5I55 
3-12440 
3-09774 
3-07155 

3-04584 
3-02057 

3-99574 
2-97135 

2-94737 
292380 

2^90063 
2^87785 

2-85545 
2-83342 
2-81175 
279043 

2-76945 

2^7488i 

2^72850 

2^70851 

2^68884 

2-66947 

2^65040 

63162 

61313 

59491 

2-57698 

55930 

2-54190 
2-52474 
2-50784 
491 19 
2-47477 

2-45859 



"I 



tangent 

-30573 

-30891 
-31210 
-31530 
•31850 
-32171 

•32492 

•32814 
•33136 
•33460 

•34108 

•34433 

-34758 
-35085 
-35412 
-35740 
•36068 

-36397 

•36727 

-37057 
•37388 
•37720 
■38053 
•38386 
-38721 
-39055 
•39391 
•39727 
•40065 

-40403 

-40741 
•4108 1 
•4 1 42 1 

•41763 
•42105 

42447 

•42791 
•43136 
■43481 
•43828 
•44175 
•44523 
secant D cotangent 



D 



318 

319 
320 
320 
321 
321 

322 

122 
324 
323 
325 
325 
325 

1^7 
2,27 
328 
328 
329 
330 
330 
331 
332 

33Z 
335 

334 

336 
338 
338 
338 
340 
340 
342 
342 
342 
344 
345 
345 
347 
347 
348 



cotangent D 



3 27085 

3-23714 
3-20406 

3-17159 
3-13972 
3-10842 

3 07768 

3-04749 
3-01783 
2-98869 
2-96004 
2-93189 
2 9042 I 
87700 
2-85023 
2-82391 
79802 
77254 
274748 
2^72281 

853 
2^67462 
2^65109 
2^62791 
2 60509 
2-58261 
2-56046 
2-53865 
2-51715 

2-49597 
2-47509 

2^45451 
2-43422 
2-41421 
2^39449 
2^37504 
235585 

2^33693 
31826 
2-29984 
2'28i67 
2-26374 
2 24604 



tangent 



I 04569 

I ^04663 
I -04757 
1-04853 
1-04950 
1-05047 

I 05146! 

1-05246 

1-05347 
1-05449 

1-05552 
1-05657 
1-05762 
1-05869 
1-05976 
1-06085 
1-06195 
I -06306 
I -06418 
I -065 3 1 
I -06645 
I -06761 
1-06878 
I -06995 
IO7115 
1-07235 
1-07356 
I -07479 
I -07602 
1-07727 

107853 

I -0798 1 
I -08 109 
1-08239 
1-08370 
1-08503 
I 08636 
I -0877 1 
I -08907 
I -09044 
1-09183 
1-09323 
I -09464 



094 

094 
096 

097 
097 
099 

100 

lOI 

102 

103 
105 
105 

107 

107 

109 
no 

1 1 1 

1 12 

113 

114 
116 
117 
117 
120 
120 
121 
123 
123 
125 
126 
128 
128 
130 
131 
133 
133 
135 
136 
-^2,7 
139 
140 
141 



cosecant D 



•95630 

•95545 
•95459 
•95372 
•95284 
•95195 
•95106 

•95015 
•94924 
•94832 
•94740 
•94646 

•94552 

-94457 
-94361 
-94264 
-94167 
-94068 

-93969 

-93869 
■93769 
•93667 

•93565 
■93462 
93358 

93253 

93148 

93042 

•92935 

-92827 

•92718 

•92609 

•92499 

•92388' 

•922761 

•92164' 

-92050 1 

-91930! 

•91822 

•91706 

•91590 

•91472 

91355 



085 

086 
,087 

I' 

1089 
091 

1091 
092 
092 

J094 
1 094 

095 

096 
097 
097 
099 

099 
100 

100 
102 
102 
103 
104 

105 

105 
106 
107 
108 
109 
109 
no 
111 
1 12 
112 
114 
114 
114 
116 
116 
118 
117 



5^ 
4C 
3< 

2C 

IC 

6£ 

5c 
4c 
3C 
20 
10 
68 
50 
40 
30 
20 

iO 

67 

50 
40 
30 
20 
10' 

66' 



D 



66^ 



20 



IT 



LOGARITHMIC FUNCTIONS 



loj; sin I D I log coscc 



•46594 

•47005 
•4741 1 
•47814 
•48213 
•48607 



401 

406 



•48998 

■49385 
•49768 



403 

399 
394 

391 
387 

3S3 
r^8i^^° 

'f'368 
f 4 36s 

''^'^ 362 
;i99i 

;235o 

2705 



jOI 

•505 

■5 

SI 



)0^^3 
50896' 



•53057 ^^8 
53405 346 



359 
355 
352 



5375^ 
•54093 
•54433 
•54769 
•55102 

•55433338 

■55761 ... 
•56085 ^^^ 
•56408 
■•56727 



342 
340 
336 
333 
33 



323 
319 
317 



317 

57044 1/ 



57978 
58284 
•58588 



306 

304 
301 
-■ ' -^ 299 
■S9I88 ^H 

.9/78 
•60070 ~ 
•60359 
•60646 

•60931 



]o^ COS 



289 

287 

285 



053406 

0-52995 
0-52589 

0-52186 

0-51787 
0-51393 

0SI002 

0-50615 
0-50232 
0-49852 

0-49477 
0-49104 

048736 

0-48371 
0-48009 
0-47650 
0-47295 
0-46944 
0-46595 
0-46249 
0-45907 
0-45567 
0-45231 
0-44898 

044567 

0-44239 
0-43915 
0-43592 
0-43273 
0-42956 

042642 

0-42331 
0-42022 
O-41716 
0-41412 
0-41 1 1 1 
0-40812 
0-405 16 
0-40222 
0-39930 
0-39641 
0-39354 
0-39069 



log SLC 



log tan D log cotan 



•48534 

•489S4 
•49430 
-49872 
•5031 1 
•50746 
•51178 
•51606 
-52031 
•52452 
•52870 
•53285 
•53697 
•54106 
•54512 
•54915 
•55315 
•55712 

•56107 

•56498 
-56887 
•57274 
•57658 
•58039 
•58418 

•58794 
•59168 
•59540 
•59909 
•60276 

•60641 

•61004 
-61364 
-61722 
-62079 
•62433 

•62785 

•63135 
•63484 
•63830 
•64175 
•64517 
•64858 



439 

435 
432 
428 

425 
421 
418 

415 

412 

409 

406 
403 
400 
397 
395 
391 
389 
3S7 
384 
381 
379 
376 



372 
369 
367 
365 



360 
358 
357 
354 
352 
350 

349 
346 
345 
342 
341 



^^^0.51466 

446 ^•5^"^^ 
442 °'5°570 
^^ 0-50128 
0-49689 
0-49254 
o 48822 

0-48394 
0-47969 
0-47548 
1 0-47 1 30 
0-46715 

o 46303 

0^45894 

o^45488 
0^45085 
0-44685 
0-44288 
043893 
0-43502 

0-43113 
0-42726 
0-42342 
0-41961 

041582 

^_^ 0-41206 
^^4 1 0.40832 
0-40460 
0-40091 
0-39724 
363:^-^9359 
0-38996 
0-38636 
0-38278 
0-37921 
0-37567 
0-37215 
0-36865 
0-36516 
0-36170 
0-35825 
0-35483 
i 0-35142 



og cotan j D 



log tan 



log sec 



001940 

0-01979 
0-02018 
0-02058 
0-02098 
0-02139 
002179 
0-0222I 
0-02262 
0-02304 
0^02347 
0-02390 



039 

039 
040 
040 
041 
040 
042 
041 
042 
043 
043 
043 



log cos 



0-024 J 7 ' 



0-02521 

0-02565 

0-02610 

0-02656 

002701 

0-02748 

0-02794! 

0-02841 

0-02889 

0-02937! 

002985 

0^03034 

0-03083 I 

0-031321 

0-03182 I 

0-03233! 

0-03283 

0-03335 

0-033861 

0-03438! 

0-03491 I 

0-03544 1 

003597 

0-03651 I 

0-03706 

0-03760 

0-03815 

0-03871 

003927 ' 

log coscc D 



044 
044 
045 
046 

045 
047 I 

0461 

047! 
048: 
048' 
048! 

049 

049 
049 
050 
051 
050 
052 

0511 
052 

053 
053 
053 
054 
055' 



054 

055 
056 

o;6 



-98060 

•98021 
-97982 
•97942 
-97902 
-97861 
-97821 
-97779 

•97738 
•97696 

•97653 
-97610 

■97567 

•97523 
•97479 
•97435 
-97390 

•97344 

•97299 

-97252 
-97206 

•97159 
•971 1 1 
•97063 
•97015 
•96966 
-96917 
-96868 
•96818 
-96767 
•96717 
-96665 
•96614 
•96562 
•96509 
•96456 
96403 
•96349 
•96294 
-96240 
-96185 
•96129 
•96073 



log sin 



66^ 



21 



24° NATURAL FUNCTIONS 



40674 

■40939 
■41204 
■41469 

•41734 
•41998 

42262 

•42525 
■42788 

•43051 
•43313 
■43575 j 
•43837 i 
•44098 i 

•44359! 
•44620 1 
•44880 
•45140 

•45399 

•45658 

•45917 
•46175 j 

•46433 
•46690 

•46947 

•47204 
•47460 
•47716 
•47971 
•48226 
•48481 

•48735 
•48989 
•49242 

•49495 
•49748 
•Soooo 

•50252 
•50503 
•50754 
•51004 
•51254 
5 1 504 



26s 

265 
265 
265 
264 
264 
263 
263 
263 
262 
262 
262 
261 
261 
261 
260 
260 
259 
259 
259 
258 
258 
257 
257 
257 
256 
256 
255 
255 
255 
254 
254 
253 
253 
253 
252 

252 



cosecant 

2^45859 

2-44264 

2-42692 

2-41142 

39614 

38107 

36620 

35154 
33708 
32282 

30875 
29487 

28117 

26766 
2-25432 
2-24116 
2-22817 

2-21535 
2-20269 

2-19019 
2-17786 
2-16568 
2-15366 
2^i4i78 

2^I3005 

2-1 1847 
2-10704 
2-09574 
2-08458 
2-07356 
2-06267 
2-05191 
2-04128 
2-03077 
2-02039 
2-OIOI4 
2-00000 
1-98998 
1-98008 
1-97029 
I -96062 
1-95106 
1-94160 
■^ secant 



990 
979 

967 
956 
946 

1) 



tangent 

44523 

44872 
45222 

45573 
45924 
46277 
46631 
46985 

47341 
47698 

48055 
48414 

48773 

49134 
49495 
49858 
50222 

50587 
S0953 

51320 
51688 
52057 
52427 
52798 

53171 

53545 
53920 
54296 
54673 
55051 
5543 1 
55812 
56194 
56577 
56962 

57348 
57735 
58124 
58513 
58905 
59297 
59691 
60086 

Dtangent 



349 

350 
351 
351 
353 
354 
354 
356 
357 
357 
359 
359 
361 
361 
363 
364 
365 
366 

367 

368 
369 
370 

Z7Z 
374 

375 
376 

378 
380 
381 
382 
383 
385 
386 
387 
389 

389 
392 
392 
394 
395 

D 



cotangent 



2-24604 

2-22857 
2-21132 
2-19430 

2-17749 
2-16090 

2-14451 

2-12832 
2-11233 
2-09654 
2-08094 
2-06553 
2-05030 
2-03526 
2-02039 
2-00569 
1-99116 
1-97681 
I -96261 
1-94858 
1-93470 
1-92098 
1-90741 
I -89400 
1-88073 
1-86760 
1-85462 
1-84177 
1-82906 
1-81649 
1-80405 
1-79174 
1-77955 
1-76749 
1-75556 
1-74375 
1-73205 
1-72047 
1-70901 
I ^69766 
1-68643 
1-67530 
I 66428 



tangent 



09464 

09606 
09750 

09895 
10041 
10189 
10338 

10488 
10640 

10793 
10947 
11103 
1 1 260 

11419 
11579 
11740 
11903 
12067 

12233 

12400 
12568 
12738 
12910 
13083 
13257 

13433 
13610 

13789 
13970 
14152 

14335 

14521 
14707 
14896 
15085 
15277 
15470 
15665 
1 5861 
16059 
16259 
16460 
16663 



osecant D 



142 

144 

145 
146 
148 
149 
150 
152 
153 
154 
156 

157 

159 

160 

161 

163 

164 

166 

167 

1< 

170 

172 

173 

174 

176 

177 

179 

181 

182 

183 

186 

186 

II 

189 

192 

193 

195 

196 

198 

200 

201 

203 



22 



24° 



LOGARITHMIC FUNCTIONS 



24° 

lO 

20 

30 
40 

50' 

25° 

10' 
20' 
30' 
40 
50' 



26° 

10' 



40 
50' 

27° 
10' 
20' 
30' 
40' 

_Jo' 

28° 
10 
20 
30 
40 
50 

29° 
10' 

20' 
30 
40' 

50' 

30° 

10' 
20 
30' 
40' 

31° 



loR 



D log cosec 



1-60931 2g3 039069 

I-6I2I4 ,,g 0-38786 
I -61494 ^-^0*38506 



1-61773 276 ^'^"^^^7 
1-62049 2-4 ^•'"'^'^ 



1-62323 
1-62595 

1-62865 
£•63133 
1-633981 



0-37951 
0-37677 

037405 



272 

270 

268°-^7i35 



265 
264 



0-36867 
0-36602 



I -63662 i^gj 0-36338 

1-63924; 260'°'^^°^^ 
1-64184^ g 0-35816 

"64442|^^^iO-35558 



•64698 



256 



1-64953!^^^ 



1-65205 1 
I -65456 1 

1-65705 

1-65952! 

1-66197 

r-66441 

1-66682 

1-66923 

1-67161 

1-67398! 

1-67633 

1-67866 

I 68098 

1-68328 

i 68557 

1-68784 
I -690 10 
1-69234 
1-69456 
1-69677 
1-69897 
1-70115 
1-70332 
1-70547 
I -70761 
1-70973 
1-71184 



2 

252 

251 

249 

247 

245 
244 
241 
241 
238 
237 
1235 
233 
232 



log; cos 



0-35302 
0-35047 

0-34795 
0-34544 
0-34295 

0-34048 
0-33803 
0-33559 
0-33318 
'0-33078 

0-32839 

10*32602 

'0-32367 

[0-32134 

0-31902 
230 "^ ^ 

^on 0-31672 

0-31443 

0-3I2I6 
0-30990 
0-30766 

0-30544 
0-30323 

0-30103 

0-29885 
0-29668 

0-29453 
0-29239 

0-29027 
0-28816 

log sec 



log tan D log cotan 



339 

338 



229 
227 

;226 

'224 

'222 
221 
220 
218 
'217 
j2I5 

:2i4 

212 
211 



1-64858 

I -65197 

1-65535 ,,, 
1-65870:^^5 

1-66204 ^^^ 
1-66537 ^^-^ 



1-66867 

I -67196 
1-67524 
1-678501 



330 
329 

328 

326 

f-68174'3^4 

r-68497 \~,\ 
i-688i8 •^"' 

- r o 320 

1-69138 I 

£-69457;^,^ 

£•69774 "f,; 
I -700891^ 
I -70404 '^j^ 
1-70717 

I -71028 1 

1-71339'^" 

i-7i648'^°9 
T4:955i307 
1-72262 
i-72567 

"72872 
73175 



log sec 



log cos 



I 

T 

£-73476 
\-7l777 
1-74077 

1-74375 

£-74673 
£-74969 
1-75264 
£•75558 
£•75852 
1-76144 

£•76435 
1-76726 
£-77015 

\j77Z^Z 
i-7759i| 

i -77877! 



307 
305 
305 
303 



!30i! 

'300 1 
298 
298 
296 
295 
294! 
294 
292 

291 

I 290 
;29ol 
288^ 
288, 
286 



03927 os6 

-03983:057 
0-04040:^^^ 
-04098.^ 

•^4156!^^ 

0-04214^58 
°'«4272 ^^ 

0-04332 

0-0439I 060 
0^04451 i 061 
0-04512 i^gj 

°-°4573'o6i! 

004634062' 

°-°4696|^62 

^•°4758o63 
0-04821 I ^ 

0-04884 i°4 

°'°4948,^6^ 

°°5oi2|o6s 

°-°5077|o65 
0-05142^6^ 

-^•^^ 207 1066 
•°5 273 1 067 
•°5340^6; 

°°S407!o67 

°-°5474lo68 
^•°5 542 1^68 

0-05679^6^ 

0-05748 I 

005818 

0-05888! 

^ 071 
0-05959 

0-06030,^^^ 

0-06102 ^^^ 

o-o6i74'^;3 

o 06247 0^3 

0-06320, 

r 074 

0-06394! '^ 

o.o6^68|^;4 

0-06543 l;^ 

0-06618 r^5 
'075 

o 06693 



log cotaii 



0-35142 

0-34803 
0-34465 
0-34130 
0-33796 
0-33463 
0-33133 
0-32804 
0-32476 
0-32150 
0-31826 
0-31503 
0-31182 
0-30862 

0-30543 
0-30226 
0-2991 1 
0-29596 
029283 
0-28972 
0-28661 
0-28352 
0-28045 
0-27738 
0-27433 
0-27128 
I0-26825 
0-26524 
0-26223 
0-25923 
025625 
0-25327 
0-25031 
0-24736 
0-24442 
0-24148 
023856 
0-23565 
0-23275 
0-22985 
0-22697 
0-22409 
0-22123 f 
log tan I log cosec D log sin 



i -96073 

T-96017 
1 1-95960 
,£•95902 

[£•95845 
'1-95786 

195728 

: £-95668 
1-95609 

1-95549 
1-95488 
1-95427 
I 95366 

£-95304 
£-95242 
£•95179 
1-95110 
1-95052 

i -94988 

1-94923 
1-94858 

£-94793 
£•94727 
I -94660 
1-94593 

£-94526 
£-94458 
£-94390 
£-94321 
1-94252 

1-94182 

T^94ii2 
1-94041 
£-93970 
£•93898 
1-93826 

i 93753 

1-93680 
1-93606 
£•93532 
£•93457 
1-93382 

i 93307 



66° 

50' 
40' 
30' 
20' 
10' 
65° 
50' 
40' 
30' 
20' 
10' 
64° 
50' 
40' 
30' 
20' 
\o' 
63° 
50' 
40' 
30' 
20' 
10' 
62° 

7o^ 

40' 

30' 

20' 

10' 

61° 

50' 

40' 

30' 

20' 

10' 

60° 

50' 

40' 

30' 

20' 

10' 

59° 



59= 



23 



31° NATURAL FUNCTIONS 



5 1 504 

51753 
52002 
52250 
52498 
52745 
52992 

53484 
53730 
53975 
54220 

S4464 

54708 
54951 
55194 
55436 
55678 

55919 

56160 
56401 
56641 
56880 
57119 
57358 
57596 
S7^U 
58070 

58307 
58543 
S8779 
59014 
59248 
59482 
59716 
59949 
60182 
60414 
60645 
60876 
61 107 
61337 
61566 



D 



249 

249 
248 
248 
247 
247 
246 
246 
246 
245 
245 
244 

244 

243 
243 
242 
242 
241 
241 
241 
240 
239 
239 
239 
238 

2Z7 
2Z7 
^17 
236 
236 
235 
234 
234 
234 

233 
232 

231 

231 
231 

230 
229 



cosecant D 



94160 

93226 
92302 
91388 
90485 
89591 
88708 

87834 
86970 
86116 
85271 
84435 
83608 
82790 
81981 
81180 



79604 
78829 

78062 

77?>o?> 
76552 
75808 

75073 
7434S 
73624 
72911 
72205 
71506 
70815 
70130 
69452 
68782 
68117 
67460 
66809 
66164 
65526 
64894 
64268 
63648 
63035 
62427 



934 

924 
914 
903 
894 
883 

874 

864 

854 
845 
836 
827 

818 

809 
801 
792 
784 
775 
767 
759 
751 
744 
735 
728 

721 

713 
706 

699 
691 
685 
678 

670 
665 

657 
651 

645 
638 

632 
626 
620 
613 

608 

n 



tangent D 
"60086 

60483 
60881 
61280 
61681 
62083 
62487 
62892 
63299 

63707 
641 17 
64528 

64941 

65355 
65771 
66189 
66608 
67028 

67451 

67875 
68301 
68728 

69157 
69588 

70021 

70455 
70891 

71329 
71769 
72211 

72654 

73100 
73547 
73996 
74447 
74900 

75355 

75812 

7(^7 i?> 
77196 

78129 

i)tanj)«nt I) 



397 

398 
399 
401 
402 
404 
405 

407 
408 
410 
411 
413 
414 
416 
418 
419 
420 
423 
424 
426 
427 
429 
431 
433 
434 
436 
438 
440 
442 
443 
446 
447 
449 
451 
453 
455 
457 
460 
461 
463 
465 
468 



cotangent 



1-66428 

1-65337 
1-64256 
1-63185 
1-62125 
I -61074 
I 60033 
1-59002 
1-57981 
1-56969 
1-55966 
1-54972 

I 53987 

I-53010 

1-52043 
I -5 1084 

1-50133 
1-49190 

1-48256 

1-47330 
I -4641 1 
I-45501 
1-44598 
1-43703 
I-42815 
I -41934 
I-41061 
I -40195 
1-39336 
1-38484 

137638 

1-36800 
1-35968 
1-35142 
1-34323 
1-33511 
I 32704 
I -3 1904 
I-31110 

1-30323 
1-29541 
1-28764 

1-27994 



tangent 



994 

985 

977 

967 
959 
951 
943 
934 
926 
919 
910 
903 
895 
888 

881 

873 
866 
859 
852 
846 
838 

832 
826 
819 
812 
807 
800 
794 
7^7 
782 

777 
770 

D 



•16C63 

-16868 
•17075 
-17283 
-17493 
-17704 
•17918 

•18133 
-18350 
-18569 
-18790 
-19012 
•19236 
•19463 
•19691 
-19920 
-20152 
-20386 
-20622 
•20859 
-21099 
-21341 
-21584 
-21830 
•22077 
-22327 
-22579 

•23089 

■2i2>A7 
•23607 

•23869 

-24134 
•24400 
•24669 
-24940 

-25214 

-25489 
•25767 
•26047 
-26330 
-26615 
I 26902 



20s 

207 
208 
210 
211 
214 

215 

217 
219 
221 
222 
224 
227 

22i 
229 
232 
234 
236 

237 

240 
242 

243 
246 
247 
250 

252 

254 
256 
251 
260 
262 
265 
266 
269 
271 
274 
275 
278 
280 

285 
287 



24 



LOGARITHMIC FUNCTIONS 



log sin 
1-71184 

£•71393 
171602 
I71809 
I72OI4 
I72218 
172421 
T72622 
172823 
173022 
I73219 
I73416 
1-73611 
173805 
173997 
1-74189 

174379 
1-74568 

1-74756 

174943 
I-75128 

I75313 
1-75496 
1-75678 

1-75859 

176039 
1-76218 
176395 
176572 
_ '6747 

i -76922 

1-77095 
1-77268 

177439 
1-77609 
177778 
1-77946 

1-78113 
T-78280 
1-78445 
1-78609 
1-78772 

I 78934 

log COS 



D log cosec 

™ 0-28816 
200 

l^^ 0-28398 

207 o 

205 °''^'?3 
,4,0.27986 

30], 0-27782 

201 1 0-27579 

,01 0-27378 

197 
197 
195 
194 
192 
192 



0-26978 

jO-26781 

1 10-26584 

026389 



0-26195 
0-26003 
0-2581 1 



;i9-362. 

^8^10-25432 



0-25244 

0-25057 
0-24872 
0-24687 
0-24504 
0-24322 
0-24141 
0-23961 
0-23782 
0-23605 
0-23428 
0-23253 
0-23078 
0-22905 
0-22732 
0-22561 



187 

185 
185 

183 
182 
181 
180 
179 
177 
177 
175 
175 
173, 

173 

1711 

17O' 

169 °-^-39i 

^^^0-22222 
0-22054 

0-21887 
0-21720 

0-21555 
0-21391 
0-21228 



167 

167 

165 

164 

163' 

162 



0-21066 

i log sec 



log tan 

177877 

178163 
1-78448 
1-78732 
I-79015 
1-79297 

i -79579 

1-79860 

r -80 1 40 
1-80419 
1-80697 
1-80975 
1-81252 
1-81528 
1-81803 
1-82078 
1-82352 
1-82626 

i'-^2899 

1-83171 
1-83442 
1-83713 
1-83984 
1-84254 
1-84523 
1-84791 
1-85059 
£•85327 
1-85594 
1-85860 

1-86126 

1-86392 
1-86656 
1-86921 
1-87185 
1-87448 
1-87711 
1-87974 
1-88236 
r-88498 
r-88759 
1-89020 

i 89281 

log cotan 



286 

285 

284 

283 

282 

282 

281 

280 

279 

278 

278 

2771 

276, 

275 I 

275 I 

274 I 

274' 

273 

272^ 

271 

271 

271 

270 

269 

2681 

268 

2681 

267 I 

266 

2661 

266 

264 

265 

264 

263 

263 

263 

262 

262 

261 

261 

261 



log cotan 

0-22123 

0-21837 
0-21552 
0-21268 
0-20985 
0-20703 
0-20421 
0-20140 
0-19860 
O-19581 

0-19303 
0-19025 

0-18748 

0-18472 
0-18197 
0-17922 
0-17648 
0-I7374 
0-17101 
0-16829 
0-16558 
0-16287 
0-16016 
0-15746 

0-15477 

0-15209 
0-14941 

0-14673 
0-14406 
0-14140 

0-13874 

0-13608 

0-I3344 
0-13079 
0-12815 
0-12552 
0-12289 
0-12026 
0-1 1764 
o-i 1502 
0-11241 
0-10980 
0-10719 
log tan 



log sec 

0-06693 

0-06770 
0-06846 
0-06923 
0-07001 
0-07079 
007158 
0-07237 
0-07317 
0-07397 
0-07478 
0-07559 
007641 
0-07723 
0-07806 
0-07889 
0-07973 
0-08058 

008143 

0-08228 
0-08314 
0-08401 
0-08488 
0-08575 



log cos 



077 

076 

077 
078 
078 
079 
079 
080 
080 
081 
081 
082 
082 

083! 
083! 
0841 
085 1 
085: 
085' 
086 1 
087! 
087! 
087! 

o,. 089 

008664 ^gg 

0-08752 
0-08842 
0-08931 
0-09022 
0-09113 
o 09204 
0-09296 

0-09389 

0-09482 
0-09576 
0-09670! 
009765 

U-0986I 

0-09957 

0-10053 
0-IOI5I 
0-10248 

0-10347 

log cosec 



090 

089 

091 

091 

091 I 

092 

093 

093; 

094 I 

094' 

095 

096 

096 

096 

098 

097 

099 



93307 

93230 
93154 
93077 
92999 
92921 
92842 

92763 
92683 
92603 
92522 
92441 

92359 

92277 

92194 
92111 
92027 
91942 

91857 

91772 
91686 

91599 
91512 

91425 
91336 

91248 
91158 
91069 
90978 
90887 

90796 

90704 
9061 1 
90518 
90424 
90330 
90235 
90139 
90043 

89947 
89849 
89752 
89653 
log sin 



52* 



25 



J" NATURAL FUNCTIONS 



•6is66 

•61795 

•62024 
•62251 

•62479 

•62706 
•62932 

•63158 
•63383 

•63608 

•63832 

■64056 

•64279 

•64501 

•64723 
-64945 
■65166 
•65386 

65606 

•65825 
■66044 
■66262 

■66480 
•66697 

66913 

67129 

67344 
67559 
67771 
679^7 

68200 

68412 
68624 

68835 

69046 
69256 
69466 

69675 
69883 

70091 
70298 

70505 
7071 1 



D 



229 

229 

227 
228 
227 

226 

226 

225 
225 

224 

224 

223 

222 

222 
222 
221 
220 
220 
219 
219 
218 
218 
217 
216 
216 

215 
215 
214 
214 
213 
212 
212 
21 I. 
211 
210 
210 
206 
208 
208 
207 
207 
206 



I 62427 

i^6i825 
1-61229 
I -60639 
I ^600 5 4 
1-59475 
I •58902 

1-58333 
I-5777I 
1-57213 
i^5666i 
i^56ii4 
I 55572 
1-55036 
1-54504 
1-53977 
1-53455 
1-52938 

I 52425 

1-51918 

1-51415 
1-50916 
1-50422 
1-49933 
1-49448 
1-48967 
I -48491 
I -48019 

I -475 5 1 
1-47087 

146628 

1-46173 
I -4572 1 
1-45274 

1-44831 
I -44391 

I 43956 

1-43524 
I -43096 
1-42672 
I -4225 1 
1-41835 
1-41421 



1602 

596 

590 
585 
579 
573 
569 
562 
I558 
552 
547 
542 

536 

'532 
1527 
522 

517 

:5i3 
507 

i503 
499 
494 
489 
485 
481 
476 
472 
468 
464 
459 
455 
452 
447 
443 
440 
435 
432 
428 
424 
421 
416 
414 



tangent 



•78129 

•78598 
-79070 

-79544 
-80020 
-80498 
-80978 
-81461 
-81946 
-82434 
-82923 
-83415 
•83910 
•84407 
•84906 
•85408 
•85912 
•86419 
•86929 
•87441 

-87955 
•88473 
•88992 
-89515 
•90040 
•90569 
•91099 
•91633 
-92170 
-92709 
•93252 

•93797 
-94345 
-94896 

-95451 
-96008 

•96569 

-97133 
-97700 
-98270 

-98843 
-99420 

00000 



J I -00000 
cotangent 



D 

469 

472 

474 
476 
478 
480 

483 

485 
488 

489 
492 
495 
497 

499 

502 

504 
507 
510 

512 

514 
518 

519 
523 
525 
529 
530 
534 
537 
539 
543 
545 
548 
551 
555 
557 
561 
564 

567 
570 
573 
577 
580 

D 



cotangent 
1^27994 

1-27230 
1-26471 
I-25717 
I -24969 
1-24227 
1-23490 
1-22758 
I -2203 I 
I-21310 
1-20593 
I-19882 
I-I9175 
I -18474 
I-I7777 
1-17085 
1-16398 
1-15715 
1-15037 

1-14363 
1-13694 
I -1 3029 
1-12369 
1-11713 
1-11061 
1-10414 
1-09770 
1-09131 
I -08496 
1-07864 

I 07237 

I -066 1 3 
1-05994 
1-05378 
I ^04766 

I ^04 1 5 8 

I 03553 

1-02952 

I-02355 
I •o 1 76 1 
I •o 1 170 
1^005 8 3 
I 00000 

tangent j 



26 



D 



764 

759 
754 
748 
742 
72>7 
732 
727 

72T 
717 
711 
707 
701 

697 
692 
687 

67S 

674 

669 
665 
660 
656 
652 

647 

644 
639 
635 
632 
627 
624 
619 
616 
612 
608 
605 
601 

597 
594 
591 
587 
583 



289 



1-2777S 
1-2807 s 
1-28374 
I 28676 

1-28980 
1^29287 

1-29597 
1-29909 
1-30223 

1-30541 

I -30861 
1-31183 
I -3 1 509 

1-31837 
1-32168 

I 32501 

1-32838 
1-33177 
1-33519 
1-33864 
1-34212 
1-34563 
1-34917 
1-35274 
1-35634 
1-35997 
1-36363 
1-36733 
1-37105 
1-37481 
i^3786o 
1^38242 
1^38628 
1-39016 

1-39409 
1-39804 
I ^40203 
I ^40606 

I^4IOI2 

I 4 142 1 



1-26902 

1-27483 295 

297 
299 
302 

304 

307 
310 
312 
314 
318 
320 
322 
326 
328 
331 
333 
337 
339 
342 
345 
348 
351 
354 

357 
360 

363 
366 
370 
372 
376 
379 
382 
386 
388 
393 
395 
399 
403 
406 

409 
D 



179- 

180 
181 

I 18. 
7 

) 



•78801 

•78622 
•78442 
•78261 
•78°79^8, 

-778971^82 
•777151,84 

•77531 ,8 

'77347 

•77162 

-76977 

-767911,8; 1 
-766041,8^ 5 

•7^417 ji88 ' 

•76229,^88 
•76041 

•75851 



185 
185 
186 



190 

190 

90 



•7566 

■7S47I 

-75280. ^^ 
-75088 9; 

-74896 ;9- 

-74509 1 ^^ K 

•74314 1 4} 



•741201 



-73924 
•7372s 

-73531 
•73333 



196 



196 

197 
198 
- 198 
•73135 8 

•72937 200 
•72737 200 
-72537 200 

-72137 
/ 00/ 201 

-72136 2^_. 

•7^934 202 

■71732 

' ' ^ 20\ 
.71529 -^ 

•71325 
-7II2I 
•70916 -^ 
-7071 1 



5( 
4^ 
3< 

2C 

4] 

5C 
4^' 
3C 

2L 
IC 

46 

5<'- 

204 40 

J04 -^^ 

:o5 
10 

45 



D 



41 



I 



LOGARITHMIC FUNCTIONS 



log sin 1 D log cosec 



178934 ,61 

1-79256 



1-79415 
1795731 
I -7973 1 



159 
158 
158 
156 
IS6 
154 
154 
153 



1-79887 

1-80043 

1-80197 

£•80351 

^•80504, 
1-806561^5- 

1-80807' 

1-80957: 

I -8 1 106 j 

1-81254I 

T-81402 [ 

T-8i549| 

1-816941 

1-81839^ 

T-81983 

1-82126 

1-82269 

1-82410 

1-82551 

1-82691 • 

1-82830I 

1-82968 I 

1-83106^ 

1-83242^ 

183378 

1-83513 
1-83648 

1-83781: 
1-83914 
1-84046 

1-84177 

T-84308 
1-84437 
1-84566 
1-84694 
1-84822 

1-84949 



151 
ISO 

149 
148 
148 
147 
145 
14s 
144 
143 
143 
141 

! 141 
140 

139 
138 
138 
136 
136 
135 
135 
133 
133 
132 
131 
131 
129 
129 
128 
128 
127 



0-21066 

0-20905 
0-20744 
0-20585 
0-20427 
0-20269 

0-20II3 

0-19957 
0-19803 
0-19649 
0-19496 

0-I9344 
0-19193 

0-19043 
|o-i8894 
0-18746 
10-18598 
:o-i845 
0-18306 
0-18161 
0-18017 
0-17874 
0-17731 
0-17590 

0-17449 

0-17309 
0-17170 
|o-i703 
10-1689 
'0-16758 
0-16622 
:o-i6487 
|o-i6352 
]0-i62i9 
jO-i6o86 
10-15954 

0-15823 

15692 
15563 
15434 
15306 
15178 
15052 



log COS I D I log sec 



log tan D 



1-89281 

I-89541 
I-89801 
T- 9006 1 
T-90320 
1-90578 

1-90837 

1-91095 

1-91353 
1-91610 
T-91868 
1-92125 
1-92381 
1-92638 
1-92894 
1-93150 
1-93406 
1-93661 
1-93916 
1-94171 
1-94426 
I -9468 1 

1-94935 
I -95 190 

1-95444 

1-95698 
1-95952 
1-96205 
1-96459 
I -967 1 2 
1-96966 
1-97219 
1-97472 
1-97725 
1-97978 
1-98231 
1-98484 
T-98737 



260 

260 
260 
259 
258 
259 
2S8 
258 
257 
258 



'log cotan 
O-IO719 

■ 0-10459 
{0-10199 

! 0-09939 
j 0-09680 
0-09422 
0-09163 



0-08905 
1 0-08647 
0-08390 
0-08132 
:% 0-07875 
^^^0-07619 



I -99242 j 
1-99495 ' 
1-99747' 
0-00000 



0-07362 
0-07106 
^^0-06850 
i ^5 0.06594 

12^. .0-06339 
0-06084 

0-05829 
0-05574 
0-05319 
0-05065 
0-04810 

o 04556 

0-04302 

0-04048 

0-03795 

0-03541 

0-032 

0-03034 

0-02781 

0-02528 

0-02275 

0-02022 

0-01769 

O-OI516 

0-01263 
2 5 2' 
•^ 10-0101 I 

253 L., 



256 
256 



255 

1255 
i255 
I254 
I255 
'254 
254 
254 
!253 
'254 

253 
'254 
253 

,253 
253 
1253 
i2S3 
■253 
253 



253 

252 

253 



10-00758 
0-00505 
0-00253 
0-00000 



log cotan I D ' log tan 



log sec 

0-10347 

0-10446 
0-10545' 
0-10646 
0-10746 
10848 
0-10950 
O-IIO52 
0-11156 
O-II259 
O-II364 
o- 1 1469 
0-II57S 
0-11681 
0-11788 
0-11895 
0-12004 
0-12113 
0-12222 
0-12332 
0-12443 
0-12554 
0-12666 
0-12779 
0-12893 
0-13007 
0-13121 
0-13237 

0-I3353 
0-13470 

0-13587 

0-13705 
0-13824 
0-13944 
0-14064 
0-14185 
0-14307 
0-14429 
0-14552 
0-14676 
0-14800 
0-14926 
0-15052 

log cosec 1 D 



099 

099, 
loi ! 
100 1 
102 
102 
102 
104 
103 
105 
105 
106 
106 
I 107 

;.o7i 

109 I 

! 1091 
1 109 

1 10 

III 
'hi; 

: 112 ! 
I 114! 

"4 

iii6l 
;ii6| 
117; 
117 
118 
119 
120 
120 
121 
122 
122 
123 
124 
1241 
126 
126 



log cos 

89653 

89554 
89455 
89354 
89254 
89152 
89050 
8S948 
88844 
88741 
88636 

88531 

88425 

88319 
88212 
88105 
87996 
87887 
87778 
87668 

87557 
87446 
87334 
87221 

87107 

86993 
86879 
86763 
86647 
86530 
86413 

86295 
86176 
86056 

85936 
85815 
85693 

85571 
85448 
85324 

85200 

85074 
84949 



45^ 



27 



FOUR-FIGURE TRIGONOMETRICAL. TABLES 



Radians 


De- 
grees 


Sine 


Cosec, 


Tangent 


Cotan. 


Secant 


Cosine 






•OOOOO 





•0000 


CO 


•0000 


00 


I -0000 


I -OOOO 


90 


1-57080 


•01745 


I 


-0175 


57-2986 


-0175 


57-2899 


I -0002 


•9998 




1-55334 


•03491 


2 


-0349 


28-6537 


-0349 


28-6362 


I -0006 


-9994 


88 


1-53589 


•05236 


3 


•0523 


19-1073 


•0524 


19-08 1 1 


1-0014 


•9986 


87 


i-5i''44 


•06981 


4 


•0698 


14-3356 


-0699 


14-3OJ6 


I -0024 


•9976 


86 


1-50-98 


•08727- 


5 


•0872 


11-4737 


-0875 


11-4301 


I -0038 


-9962 


85 


I -48 - - 


•10472 


6 


•1045 


9-5668 


•IO51 


9-5144 


1-0055 


•9945 


84 


i-46(.;' 


•12217 


7 


•1219 


8-2055 


•1228 


8-1443 


1-0075 


•9925 


83 


1-44^^:. 


•13963 


8 


•1392 


7-1853 


•1405 


7-1154 


I -0098 


•9903 


82 


1-431^7 


•15708 


9 


•1564 


6-3925 


•1584 


6-3138 


1-0125 


•9877 


81 


1-41372 


•17453 


10 


•1736 


5-7588 


•1763 


5-6713 


I-OI54 


•9848 


80 


1-39626 


•19199 


II 


•1908 


5 '2408 


•1944 


5-1446 


I-O187 


.9816 


79 


1-77881 


•20944 


12 


•2079 


4-8097 


•2126 


4-7046 


1-0223 


•97 I 


78 


I--6i::5 


•22689 


13 
14 


•2250 
•2419 


4-4454 
4-1336 


•2309 
-2493 


4-3315 


1-0263 


•972 


77^ 
76 


J -34:90 


•24435 


4-0108 


I -0306 


-9703 


1-3264 


•26180 


IS 


•2588 


3-8637 


-2679 


Vll^-i^ 


1-0353 


-965^ 


75 


i'3(^<-. 


•27925 


16 


-2756 


3-6280 


•2867 


3-4874 


I -0403 


-961 


4 


1-291-- 


•29671 


17 


•2924 


3-4203 


•3057 


3-2709 


1-0457 


-9563 


73 


"r274c 


•31416 


18 


•3090 


3-2361 


•3249 


yo777 


I-0515 


-95 II 


72 


I..56C 


•33161 


19 


.3256 


3-0716 


•3443 


2-9042 


1-0576 


•9455 


71 


|.239ifc 


•34907 


20 


-3420 


2-9238 


•3640 


2-7475 


I -0642 


-9397 


10 


1-22173 


•36652 


21 


•3584 


2-7904 


•3839 


2-6051 


I -07 1 1 


-9336 


*/%- 


1-20428 


•38397 


22 


-3746 


2-6695 


•4040 


2-4751 


1-0785 


•9272 


^9 


r-18682 


•40143 


23 


•3907 


2-5593 


-4245 


2-3559 


I -0864 


•9205 


ft? 


/-16937 


•41888 


24 


•4067 


2-4586 


-4452 


2-2460 


I -0946 


-9135 


0^ 


/•15192 


•43633 


25 


•4226 


2-3662 


•4663 


2-1445 


I-IO34 


-9063 


65 


1-13446 


•45379 


26 


•4384 


• 2-2812 


•4877 


2-0503 


I-II26 


•8988 


64 


1-11701 


•47124 


27 


-4540 


2-2027 


-5095 


1-9626 


I-1223 


•8910 


63 


1-09956 


•48869 


28 


•4695 


2-1301 


-5317 


1-8807 


1-1326 


-8829 


62 


I -082 10 


•50615 


29 


•4848 


2-0627 


•5543 


I -8040 


I -1434 


•8746 


61 


I -06465 


•52360 


30 


•5000 


2 -0000 


•5774 


1-7321 


1-1547 


•8660 


60 


I -04720 


•54105 


31 


•5150 


I -9416 


•6009 


I -6643 


I-I666 


-8572 


59 


1-02974 


•55851 


32 


•5299 


1-8871 


•6249 


I--6003 


1-1792 


-8480 


S8 


1-01229 


■57596 


33 


•5446 


1-8361 


•6494 


1-5399 


i^i924 


•8387 


57 


-99484 


•59341 


34 


•5592 


1-7883 


•6745 


1-4826 


1-2 2 


■8290 


56 


•97738 


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35 


-5736 


1-7434 


•7002 


1-4281 


1-2^08 


-8192 


55 


-95993 


•62832 


36 


-5878 


1-7013 


•7265 


1-3764 


1-236; 


•8090 


54 


-94248 


•64577 


37 


•6018 


I -6616 


•7536 


1-3270 


I-2521 


•7986 


53 


•92502 


•66323 


38 


-6157 


1-6243 


•7813 


1-2799 


I -2690 


•7880 


52 


-9075; 


•68068 


39 


•6293 


1-5890 


•8098 


1-2 349 


1-2868 


-7771 


51 


-8901 


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40 


•6428 


1-5557 


•8391 


1-1918 


1-3054 


-7660 


50 


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41 


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1-5243 


•8693 


1-1504 


1-3250 


•7547 


49 


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•73304 


42 


•6691 


I -4945 


•9004 


1-1106 


1-3456 


-7431 


48 


•83776 


•75049 


43 


•6820 


1-4663 


•9325 


1-0724 


1-3673 


-7314 


47 


•82030 


•76794 


44 


•6947 


1-4396 


•9657 


I-0355 


1-3902 


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46 


•80285 


•78540 


45 


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1-4142 


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45 

De- 
grees 


•78540 






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Cotan. 


Tangent 


Cosec. 


Sine 


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28 



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•38 
•40 
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•45 
•47 
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•61C 

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•7 If 
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•767 
.785 



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OCT 5 'bJ-J^ 



LD 21A-50m-3,'62 



General Library 

University of California 

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UNIVERSITY OF GAUFORNIA LIBRARY 



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