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AN INTRODUCTION TO
THE STUDY OF
NUMERICAL TRIGONOMETRY
Cambridge University Press
Fetter Lane, London
Nc e w York
Bombay, Calcutta, Madras
Toronto
Macmillan
Tokyo
Maruzen Company, Ltd
All rights reserved
AN INTRODUCTION TO
THE STUDY OF
NUMERICAL TRIGONOMETRY
by
E. A. PRICE, B.A.
Assistant Master at the Royal Naval College, Dartmouth
Formerly Assistant Master at Winchester College
Cambridge :
at the University Press
First Edition 11)12
licpnnted iDltt, 1021, l'.)24, 1930
PK1NTLJ) IN OllKAT lUUTAIN'
PREFACE
IN reprinting this little work, which formerly appeared
under the title of Examples in Numerical Trigonometry,
some slight alterations and additions have been admitted. .
The book is an attempt to introduce the elements of
Trigonometry on heuristic lines, and is intended for those
pupils who do not intend to pursue the study of Mathematics
very far. It has for some time been an accepted principle
that for such pupils a wider range of subjects treated in a
less formal fashion should take the place of that drill in the
manipulation of symbols which is essential for the specialist
in Mathematics. The author has endeavoured to carry out
this principle, and has to a large extent omitted formal
proofs and deductions the theory of the machine which it
is proposed to handle.
Even the slight acquaintance with this branch of Mathe-
matics which is aimed at here, will to a large extent miss its
true opportunity unless some notion of "functionality" or the
growth of variable quantities, is acquired, and with this end
in view a graphical treatment of the Ratios is introduced,
the same idea being applied in dealing with the ratios of the
Obtuse Angle.
Among the new features it will be found that headings
have been added to the various sections ; and the preliminary
drawing exercises have been reduced and questions substi-
tuted which are intended to call into play the "intuitive".
VI PREFACE
faculty the ability to deduce the general rule from few
concrete examples by the exercise of the imagination, with,
of course, subsequent verification. Additions have also been
made to Chapter X.
A chapter on Logarithms has been introduced before the
general treatment of triangles, but it should be noted that
their use is by no means imperative.
Thanks are due to several friends who have kindly
assisted in the working out of the answers, and especially to
the Director of Naval Education and the Headmaster for
permission to make use of questions set in the Examination
papers at Osborne.
E. A. P.
January 1918
CONTENTS
CHAP. PAOE
L Measurement of Angles. Angles of Elevation and
Depression. Subtended Angles. Bearings . . 1
II. The Tangent and its use in Problems .... 6
III. The Sine and its use in Problems .... 13
IV. The Cosine and its use in Problems . . , . 17
V. The Secant, Cosecant, and Cotangent .... 20
VI. Miscellaneous Problems involving right-angled triangles 24
VII. Areas 35
VIII. Logarithms 38
IX Solution of triangles which are not right-angled . . 48
X. Ratios of Obtuse Angles 53
XL The Sine Formula 55
XII. The Cosine Formula 58
XIII Problems 61
APPENDICES.
Definitions of the Trigonometrical Ratios and relations
between them 72
Proof of the Sine Formula 74
Proof of the Cosine Formula ..... 76
Examples of orderly arrangement of a Problem , . 76
ANSWERS , 79
CHAPTER I
MEASUREMENT OF ANGLES.
If two straight lines cut one another in such a way that all
the angles between them are equal, then each of these angles is
called a right anyle.
A right angle is divided into 90 equal degrees. (90)
A degree is divided into 60 equal minutes. (60')
A minute is divided into 60 equal seconds. (60")
[A second is such a small angle that it is used only in very accurate
calculations, e.g. in Astronomy. In this book only Degrees and Minutes
n*e used to measure angles.]
1. How many degrees are there in 2 right angles, J right
angle, ^ right angle, T ^ right angle, 0'3 right angle?
2. How many degrees arid minutes in ^ right angle, y 1 ^ right
angle, f right angle, right angle (to nearest minute) ?
3. Find the sum of each of the following pairs of angles :
(1) 22 10' (2) 29 25' (3) 38 51' (4) 79 57'
31 15' 47 35' 19 43' 59 49'
4. If two angles together make up a right angle they are
said to be complementary, and each is called the complement of
the other..
Calculate the complements of each of the angles in question 3.
5. The three angles of a triangle are together equal to two
right angles. If each pair in question 3 represents two angles of
a triangle, what is the third angle in each case 1
1
MEASUREMENT OF ANGLES
[CH.
6. If ABC is a right-angled triangle with C a right angle
and CD drawn perpendicular to AB, calculate the number of
degrees in every acute angle of the figure, and insert the results
in a small sketch when A contains (1) 30, (2) 67, (3) 43 21'.
7. Find the number of degrees in each angle of an isosceles
triangle if each of the base angles is (1) twice, (2) four times,
(3) half the vertical angle. Insert answers in a sketch.
Note. In the two annexed figures the angle ACB is said to be subtended
at C by the line (or arc) AB, and the angle APB is the angle subtended by
AB at P.
Fig. 1.
Fig. a.
8. How many degrees are there in the angle at the centre
of a regular 5-sided figure subtended by one of the sides 1
How many degrees are there in each of the interior angles,
and in each of the exterior angles of the 5-sided figure ?
9. Repeat question 8 for (1) an 8-sided, (2) a 9-sided,
(3) a 7-sided regular figure.
Note. In recording the bearing of one point from another the angle is
always measured from a meridian (i.e. a North and South line). If a ship
lies N. 31 E. of a lighthouse, a man on the lighthouse will find it by
looking first due North and then turning through an angle of 31 to the
East.
10. If A is N. 31 E. of B, what is the bearing of B from A 1
11. If P is N. 39 E. of Q, and R is S. 51 E. of Q, what
angle does PR subtend at Q)
I] MEASUREMENT OF ANGLES 3
12. If X is N. 21 W. of Y, and Z is N. 43 E. of X, and if
XY subtends an angle of 30 at Z, what is the bearing of Z
from Y 1
13. Q is N. 60 E. of R, P is S. 10 E. of a If RQ subtends
an anglo of 35 at P, what is the bearing of P from R?
Note. From a boat B a man is observed at the top of a cliff AC. The
angle ABC is said to be the angle of Elevation of the man as seen from the
boat, and the angle BAD is said to be the angle of Depression of the boat as
seen from the top of the cliff.
Tig. 8.
Remember that angles of Elevation and Depression are always measured
from the Horizontal.
14. There is a church-tower surmounted by a spire. If the
angle of elevation of the top of the spire as seen from a certain
point on the ground is 53, and the spire subtends an angle of
29 at the same point, what is the angle of elevation of the top
of the tower from the same point?
15. From a boat at sea the angle of elevation of the top of
a certain cliff is 37. What is the angle of depression of the
boat as seen from the top of the cliff?
12
4> MEASUREMENT OF ANGLES [CH. I
16. From the masthead of a ship the angles of depression of
two buoys one behind the other are found to be 19 and 24.
What angle does the line joining them subtend at the masthead?
17. What angle does one-sixth of the circumference of a
circle subtend (1) at the centre, (2) at any point on the circum-
ference]
18. What fraction of the circumference is subtended by
angles of 120, 60, 20, at points on the circumference?
19. If in Fig. 1 above C = 90% L. CPB = 73 and L. BAG = 25,
how many degrees are there in each of the angles PBC, ABC, ABP?
20. If in the same figure C = 90, L. CPB = x* and L BAG = y,
how many degrees are there in u ABPl
21. If the edges of the square pyramid in Fig. 8 are each
10 cm. long, how many degrees are there in each of the angles
ACB, ACD, BCD, CAK?
Is L AKN larger or smaller than L ACN ?
If K were moved along DC towards C, would the angle AKN
increase or decrease ?
22. From a ship sailing parallel to a straight coastline the
angle subtended by the line joining two iixed points on shore is
observed. Does this angle vary in size? (Draw a sketch.) If
so, when is it greatest ?
23. If the angle observed in the last question remained the
same for any length of time, would the ship be still moving
parallel to the shore? If not, describe carefully her course
during that time.
CHAPTER II
THE TANGENT AND ITS USE IN PROBLEMS
Instruments, squared paper, and Table of Natural Tangents
required.
Note. Express all results in decimals to three significant figures.
1. Drawing accurately with
instruments construct an angle
Q of 59*. Take a point P in four
different positions somewhere on
one arm of the angle, and draw
PN perpendicular to the other arm.
Measure PN and NQ in inches and
decimals of an inch, and calculate
PN .
the value of the fraction in deci-
NQ
mals for all four positions. Arrange
in a table thus:
Fig. 4.
1.
2.
3.
4.
PN
What do you observe with regard to these values of ]
NGl
2. Repeat question 1, making Q equal to 39.
* The angle Q in the diagram is purposely drawn incorrectly to avoid
being copied.
6
THE TANGENT AND
[cn.
3. If another angle (20 say) were drawn in the same way,
would you obtain a similar result ?
What do you observe with regard to the value of the fraction
PN
for each angle Q? Express clearly in words and remember
the result.
PN
4. The fraction (or ratio) - is called the Tangent of the
NQ
angle at Q, and from question 1 is obtained : tan 59 = 1-66.
Write down the result of question 2 in the same way.
5. By drawing accurate figures, find the values of tan 47,
tan 29-5, tan 34 24' as exactly as possible.
6. Copy the annexed figure ac-
curately, making QN = 5cm., PNQ=90
and the angles at Q each 10, the
whole angle at Q being 60.
Measure the length of PN when
the angle Q is 10, 20, 30, 40, 50,
60, and calculate the tangent of each
of these angles.
7 Turn to the Table of Natural
Tangents and search for the results
of the last question. Make a com-
parative table showing the results
obtained by drawing and from the
printed Tables thus:
Fig. 5.
Angle
Tang
by Drawing
;ent
from Tables
10
0-1763
20
0-3640
30
etc.
etc.
II] ITS USE IN PROBLEMS 7
8. Make a similar table showing the results of questions 1,
2, and 5 above.
Note that the values given in the Tables are correct to four significant
figures only. Tan 20 = 0-3639702 correct to seven figures, but this is not
absolutely exact. Four significant figures give results sufficiently accurate
for ordinary calculations.
9. Plot on squared paper the values obtained in question 7,
i.e. draw the graph of y tan x for values of x between and
60. What are the tangents of 0, 45, 90 ?
Write down the following sentence and insert the word
increases or decreases, whichever is appropriate :
"As the angle x increases from to 90, tana? "
10. Draw A ABC having C = 90, AC = 3 cm., BC - 4 cm.
Measure AB in cm. and the angles A and B in degrees and
minutes as accurately as possible.
Write down (in decimals to three significant figures) the values
of the tangents of the angles A and B as found from the figure.
11. If drawn very accurately, the angle A in the last question
should he 53 8', and tan 53 8' = 1-3335. Search for this in
the Table of Natural Tangents and write down the values of
tan 53 6' and tan 53 12'.
12. The angle B in questioi 10 should have been 36 52', and
tan 36 52' = 0*7499. Search this out and write down the values
of tan 36 48' and tan 36 54'.
13. Write down each of the following and find its value
from the Tables: tan 56, tan 56 24', tan 56 30', tan 56 27',
tan 28, tan 28 12', tan 28 18', tan 28 15', tan 16 25', tan 73 22',
tan 3 51'.
14. Is the tangent of an angle half the tangent of twice the
angle? Compare tan 40 and tan 80 and draw a freehand sketch
to illustrate your answer. (See figure of question 6.)
15. From the Tables find the angles whoso tangents are
0-4663, 3-0777, 0-3153, 1-4388, 0-4023, 3-6888, 0-1974, 2-3164,
1-1980, 0-6633.
8
THE TANGENT AND
[CH.
16. If an angle X is such that tan X - 0-7 (or ^) it may be
constructed by drawing a right angle included between two lines,
one of which is seven-tenths of the other. Draw such a triangle
having sides 7 cm. and 10 cm., and measure the angle whose
tangent is 0'7. Verify the result hy reference to the Tables.
17. Find in the same way the angles whose tangents are
(1) 0-8, (2) -2-0, (3) 3*5. Verify from the Tables.
EASY PROBLEMS
18. From a distance of 100 feet measured along level ground
from the foot of a vertical tower the angle of elevation (see p. 3)
of the top is found to be 29. Find by drawing a small freehand
sketch and using the Table of Natural Tangents what the height
of the tower must be.
19. The angle of elevation of the top of a flagstaff from a
point 40 feet from its base is 50. Calculate its height.
20. If a ladder is placed with its base 5 feet from the
bottom of a wall 12 feet high and just reaches the top, what
angles does the ladder make with the ground and the wall, and
how long must it be 1
21. A ship sails 5 miles due West and then 7 miles due
North. When it is in this position calculate the distance and
bearing of the point from which she started.
22. If the angle of elevation of a monument from a point
200 feet from its base and on the same level is 34, what is the
height of the monument ?
23. A vertical pole 12 feet high casts a shadow 20 feet long.
What must the angle of elevation of the sun be?
shadow
Fig. 6.
Il] ITS USE IN PROBLEMS 9
RIGHT-ANGLED TRIANGLES
Note. In a triangle ABC the sides opposite the angles A, B, C are usually
called a, ft, c.
All units of length in the following examples are centimetres. All lengths
should be given to four significant figures, all angles in degrees and minutes.
In the triangle ABC having C = 90
24. Find b if B = 27 and a = 10 units of length.
25. Find b if B = 27 12' and a - 10.
26. Find b if B = 27 16' and a= 10.
27. Find b if B = 27 16' and o = 2-5.
28. Find a if A = 49 28' and b = 2-3.
29. Find a if A = 84 24' and b = 12-5.
30. Find A if a = 4 and b = 5.
31. Kind A and B if a = 8 and b = 10.
32. Find A and B if a = 2-4 and 6=1-5
33. Find A and B if a = 13'6 and b = 19'9.
SOLUTION OF A TRIANGLE
Note. To solve a triangle when three of its sides and angles are given
means to calculate the remaining sides and angles.
Solve the triangle ABC if C = 90 and
34. A = 42 17' and 6 = 25 -4.
35. B = 29 11' and a =1-62.
36 A =13 53' and a = 10-4.
37. B = 72 41' and b -4-07.
38. a = 12-4 and b = 9-3.
39. a = 124-5 and 6 = 150.
PROBLEMS
40. From a point 27 feet from the wall of a building a
window sill is observed, and its angle of elevation found to bo
63 26'. Find its height above the ground to the nearest inch.
10 THE TANGENT AND [OH,
41. To find the range of a distant object C, two men A and B
take up positions BO that AC is at right angles to AB. If AB is
20 yards, and the angle ABC is found to be 85*4, find AC.
42. A thin stratum of gold-bearing rock crops out on the
level surface and is known to slope down at an angle of 7 13',
If a shaft is driven vertically down at a distance of 1J miles
from the outcrop, at what depth in feet will it meet the stratum?
43. In order to measure the width of a river a distance AB
is marked out straight along the bank; a point C is observed on
the bank exactly opposite to A and the angle CBA is measured.
If AB50 yards and CBA = 54 25', find the width of the
river.
44. The rails approaching a tunnel under a certain estuary
slope down at an angle of 5 37' and are at water level at a
distance of 1700 feet from the water's edge. At what depth (in
feet) below the surface of the water are they when exactly under
the water's edge ?
45. In a triangle ABC having O = 90, AC = 5 inches and
BC = 12 inches, BD is drawn at right angles to A B to meet AC
produced in D. Find the length of CD.
46. A chord 6 cm. long is 2 cm. from the centre ot a circle.
What angle does it subtend (see p. 2) at the centre ?
47. If a chord 10 cm. long subtends 140 at the centre of a
circle, how far must it be from the centre?
48. From a point on level ground 75 feet from the base of a
flagstaff, the angles of elevation of the top and of the point at
which a yardarm is fixed are found to be 49 37' and 38 29'.
Find the height of the top and of the yardarm above ground.
49. On a map a piece of straight road between the 100-foot
and 400-foot contours seems to be J mile long. What is its
slope and how many yards long is it really?
50. The poles for a certain Wireless Telegraphy Station con-
sist of three parts fixed vertically one above the other, each part
II] ITS USB IN PROBLEMS 11
being 50 feet long. The whole is held erect by wire stays fixed
to the top of each part and to points in the ground 75 feet away
from the base. Find the angles which each of the wire stays
makes with the ground and with the poles. Find also the
angles between the three wire stays fixed to the same point on
the ground. Draw a sketch and insert the answers,
51. From a ship sailing up a river the angle of elevation
of a point on a bridge 176 feet above the water and straight in
front is found to be 29 14'.
If the ship is moving at a rate of 5 miles an hour, in how many
seconds will it be exactly under the bridge?
52. The figure is a sketch for the following problem :
200 Art
Pig. 7.
From a boat the angle of elevation of an object on the top of
a cliff 200 feet high is found to be 29. The boat sails directly
towards the object, and after a certain interval the angle of
elevation is found to be 50. How many feet did the boat sail!
[Calculate L ACD, L BCD, AD, BD ; then AB can be found.]
53. Repeat question 52, making L A = 27 13', CD = 274 feet
and L CBD = 63 29'.
*
54. The angle of elevation of the top of a tower is observed
to be 25, and from a point 100 feet nearer to the foot of the
tower it is 43. Find the height of the tower.
12
THE TANGENT AND ITS USE IN PROBLEMS [CH. II
55. Repeat question 52, with angles 18 47' and 51' 29' and
distance 124 feet.
56. The figure represents a square pyramid each of whose
edges is 10 cm. long. It is required to find the angle (ABN)
A
Fig. 8.
between one slant-edge and the base, and also the angle (AKN)
between one face and the base.
Note that ANB, ANC, ANK, AND are right angles. Calculate BN and
AN by Pythagoras' Theorem.
57. Calculate the height of a regular tetrahedron (triangular
pyramid) whose edges are each 10 cm. long. Find also the angles
between slant-edge and base, and between two faces.
Note NK = JBK.
A
Fig. 9.
CHAPTER III
THE SINE AND ITS USE IN PROBLEMS
Instruments, squared paper , and Table of Natural Sines
required.
1. Construct an angle Q of 59 carefully with instruments.
Take a point P in four different positions somewhere on one arm
of the angle and draw PN perpendicular to the other arm.
Measure PN and PQ for each position of P and calculate the
PN
value of the fraction in each case. Tabulate the values of
PN
PN, PQ, and . [Drawings for Chap. IT may be used.]
2. Ilepeat question 1, making Q = 39.
PN
3. What do you observe with regard to the fraction -- in
each case? Express clearly in words and remember the result.
PN
4. The fraction (or ratio) is
PQ
called the Sine of the angle GL What
is the value of sin 59 t
Write down the result of ques-
tion 2 in the same way.
5. By drawing accurate figures,
find the values of sin 47, sin 2 9 -5,
sin 34 24' as exactly as possible.
6. Copy the annexed figure ac-
curately, making the radius of the
circle 10 cm. long and putting in
radii at intervals of 10.
14
THE SINE AND
[CH.
If P be the end of a radius and PN be drawn perpendicular
to NQ, then by measuring PN in all positions the sines of the
angles 10, 20, 30, 40, 50, 60, 70, 80 can be calculated.
Find these values and make a table. What is the value
of sin and of sin 90 ?
7. Plot these values on squared paper (i.e. draw the graph
of y = sin x for values of x between and 90). From the graph
find the values of sin 47 and sin 29*5, and compare with the
results of question 5. Does sin x increase or decrease as x
increases from to 90 ? Answer quite clearly.
8. Compare the results obtained in question 6 with the values
given in the Table of Natural Sines. Arrange in a table thus :
Angle
Si
by Drawing
ne
from Tables
o-oooo
10
0-1736
20
0-3420
etc.
etc.
9. In Ex. 10, Chap, n a triangle ABC having = 90,
AC = 3 cm. and BC = 4 cm. was drawn, and it was found that
AB = 5 cm. and A = 53 about. Draw a neat freehand sketch of
this triangle. If drawn very accurately the angle A should be
53 8', and sin 53 8' = 0*8000 (or |). Search for this in the Table
of Natural Sines and write down the value of sin 53 6', and of
sin 53 12'.
10. The angle B in the triangle above should be 36 52', and
sin 36 52' = 0-5999 (or nearly).
Search for this in the Table of Natural Sines, and write down
the value of sin 36 48' and of sin 36 54'.
Ill] ITS USE IN PROBLEMS 15
11. Write down each of the following and find its value
from the Tables :
sin 35 12', sin 35 18', sin 35 15', sin 35 13',
sin 72 55', sin 43 9', sin 10 53', sin 81 49'.
12. Is the sine of an angle half the sine of twice the angle ?
Compare sin 40 and sin 80 and draw a freehand sketch to
illustrate your answer. (See figure of question 6.)
13. From the Tables find the angles whose sines are
0-4226, 0-9455, 0-8141, 0-2890,
0-2898, 0-9795, 0-9701, 0-7202.
14. By drawing a right-angled triangle with one side 5 cni.
long and the hypotenuse 10 cm. long and measuring the angles,
find the angle whose sine is *5 (= T 6 ^). Verify from the Tables.
15. Find in the same way the angles whose sines are (1) 0-4,
(2) 0-75, (3) 0-43. Verify from the Tables.
RIGHT-ANGLED TRIANGLES
16. If in A ABC having C = 90 (draw freehand sketch)
(1) B = 27, c= 10 units, find b.
(2) B = 2712', c = 10, find b.
(3) B = 271G', c = 10, find b.
(4) B = 27 16', c = 2-5, find b.
(5) A = 43 37', c = 12-2, find a.
(6) A = ll22', c=l 1-22, find a.
17. In the triangle ABC, C = 90, B = 63 28' and o - 75 cm.
Find b, a (by Pythagoras) and A. Verify by drawing to scale.
18. Solve the triangle ABC if C = 90 and
(1) A = 72 14', c = 21. (Do not use Pythag.)
(2) B = 2523', c-10-4.
(3) B = 4213', c = 105-2.
(4) A=ll19', c=19-32.
16 T11E SINK AND ITS USE IN PROBLEMS [CH. Ill
PROBLEMS
19. A straight road rises 48 feet in 400 yards measured
along the road. Find its inclination to the horizontal.
20. What is the angular slope of a railway lino whose
gradient is 1 in 100? (The 100 is measured along the rails.)
21. What angular slopes are represented by the boards at
the side of a railway line which
are shown in the figure 1
22. A ladder 25 feet long
is placed against a wall and
makes an an^le of 73 22' with
the ground. To what height Fig. 12.
does it reach? Answer to nearest inch.
23. What is the height of an isosceles triangle whose equal
sides are each 2-5 inches long and whose base angles are 17 35'
each?
24. A lighthouse 13 miles away is observed from a ship to
be 19 off the course of the ship. If the ship continues on her
course, how far from the lighthouse will she be when she is
nearest to it?
25. A rod 15 inches long hangs from a nail in the wall. If
it is pulled away so that it makes an angle of 25 33' with the
wall, what is the distance of the lower end from the wall ]
26. What angle will the rod in the last question make with
the vertical when its lower end is 8 inches from the wall ?
27. A trap-door is 3 feet 3 ins. wide from hinge to opposite
edge. If it is raised through four-fifths of a right angle, how
high above the ground will the outer edge be ?
28. A framework of four rods each 1 foot long, hinged at the
ends, and with elastic diagonals, is laid upon the table. Find
the lengths of the diagonals when the angle between two adjacent
sides is (1) 45 22', (2) 145 22'.
CHAPTER IV
THE COSINE AND ITS USE IN PROBLEMS
Definition. If from any point P in one arm of an angle Q a
perpendicular PN be drawn to the
Q,N
other arm, the ratio is called
the cosine of the angle Q.
Kemember this definition.
1. Draw any acute angle ac-
curately with instruments and show
by careful measurements that the
cosine of this angle remains the same for any four different
positions of the point P. Arrange results in a table.
2. Find by accurate drawing the value of cos 59 and of
cos 39 (cos 59 means the cosine of an angle of 59).
3. Find by accurate drawing the angles whose cosines are
(1) 0-5, (2) 075, (3) 0-36.
4. Copy the annexed figure ac-
^urately making the radius 10 cm.
long and putting in radii at intervals
of 10.
By drawing PN in all positions
and measuring NQ, find the cosines
of the angles 0, 10, 20, 30, ... 90.
Tabulate the resulting values.
Does the cosine increase or de-
crease as the angle increases from
to 90 7
Fig. 14.
P. T.
13 THE COSINE AND [OH.
5. Compare the results obtained in question 4 with the
values given in the Table of Natural Cosines.
To the table of results in that question add a column showing
the values given in the printed Table.
6. From the Table of Natural Cosines find and write dowr
the values of cos 56, cos 56 24', cos 56 30'.
Now cos 56 27' lies between the last two values. Find it,
and write it down.
7. Find from the Tables the values of
cos 28 15', cos 16 25', cos 73 21',
cos 81 51', cos 53 8', cos 12 28'.
8. In a triangle ABC having C = 90 and
(1) A = 27, c=10, find*.
(2) A = 43 22', c=10, find b.
(3) B = 19 25', c= 8-8, find a.
(4) B = 85 17', 0=12-2, find a.
PROBLEMS
9. A wire 100 feet long is stretched from the top of a flagstaff
to a point in the ground. If it makes an angle of C3 29' with
the ground, how far is this point from the foot of the flagstaff?
10. -A ship has sailed North- West for 21 miles. How far
North of her starting-point is she ?
11. A straight road running uphill at an angle of 9 35' with
the horizontal is 1 mile long. How many yards long will it
appear to be on the map?
12. A ladder 25 feet long leans against a wall making an
angle of 59 43' with the ground. How far is its foot from the
foot of the wall ?
13. What is the length of the base of an isosceles triangle
whose equal sides are 2 '5 inches long and whose base-angles are
each 1734'f
IV]
ITS USE IN PROBLEMS
19
14. Gloucester is 79 miles N. 29 W. of Southampton. How
far N. of Southampton is Gloucester?
15. A rod 3 feet long hangs from the ceiling. How far is
the lower end from the ceiling when the rod makes an angle of
69 26' with the vertical?
16. What is the length of the shortest side of a set-square
with angles of 90, 60, 30, if the longest side is 15 cm. long?
17. A flagstaff whose top when vertical is 70 feet above the
ground is found to lean to one side making an angle of 82 20'
with the ground.
What is the length of its shadow when the sun is vertically
overhead ?
18. A roof is made of pieces of corrugated iron leaning
against a wall and supported at their lower onds by poles. If
the length of each piece of iron is 6 feet 6 inches, and it makes
an angle of 32 with the horizontal, what width will be kept dry
when rain falls vertically ?
19. If the latitude of New York be taken as 40 N. and the
radius of the Earth as 4000 miles,
calculate the distance of New York
from the axis NS of the Earth. Through
what distance does New York revolve
each day? (See Fig. 15.)
20. How many miles a minute is
London moving through space owing
to the spin of the Earth on its axis ?
Take London's latitude as 51 30'.
21. What is the distance round
the world in latitude 60? On Mer-
cator's Projection of the World the
width of the Atlantic Ocean in latitude 60* N. appears to be the
same as its width on the Equator.
What is the true relation between these distances ?
22
CHAPTER Y
THE SECANT, COSECANT, AND COTANGENT
Note. These ratios are shortened to sec A, cosec A, cot A.
1. Find the value of sec 36 from the Table of Natural
Secants. Find also the value of cos 36 u Multiply these
together and give the answer correct to four significant figures.
2. Repeat twice with any other angles.
What do you observe with regard to the secant and cosine
of any one of these angles 1
Note. } x f = 1, and J is called the reciprocal of } .
3. Find in a similar way some relation between the cosecant
and some other ratio known already.
4. Repeat for the cotangent and tabulate the relations found
in the first four questions. The results should be remembered.
5. If ABC is a triangle
with C = 90, express cosec A,
sec A and cos A each as a ratio
of two sides.
6. Find from the Tables
and write down the values of
sec A, cosec A and cot A when
A is
56, 56 24', 56 30', 56 27';
28, 28 12', 28 18', 28 14';
16 25', 73 21'.
CH. V] THE SECANT, COSECANT, AND COTANGENT 21
7. Write down the following and complete it by inserting
the words " increases" and "decreases" :
" As the angle A increases from to 90, sec A..., cosecA...,
cot A...." Illustrate with a freehand drawing.
PROBLEMS
Note. Use secants in questions 8 to 11.
8. A wire is stretched from the top of a vertical pole to
a point in the ground 20 feet from the base. How long must
the wire be if it makes with the ground an angle of (1) 28,
(2) 53 19', (3) 61 58'?
9. The straight line joining the tops of two vertical poles
31 feet apart is found to make an angle of 19 23' with the
horizontal. What is the distance between the tops?
10. From the top of a cliff 470 feet high the angle of
depression of a boat is found to be 23 14'. How far is the
boat from the top of the cliff? [Find the complement.]
11. Find the radius of the circle circumscribing a regular
five-sided figure whose sides are each 12 cm. long. [Calculate
the angle at each corner.]
Note. Use cotangents in questions 12 to 16.
12. A ladder makes an angle of 41 26' with a vertical wall.
If its lower end is 10 feet from the base of the wall, to what
height does the top reach?
13. One side of a rectangle is 15 inches long and the
diagonal makes an angle of 17 23' with the other side. Find
the length of the other side.
14. How far from a tower 110 feet high must a man be
standing if the angle of elevation of the top is 35 4'!
15. A is 25 miles due N. of B, arid C is due E. of B. If AB
makes an angle of 13 58' with AC, Hud the angle C and BC.
22 TUB SECANT, COSECANT, [CH.
16. P is a point on the circumference of a circle whose
diameter is XY. If the straight line from P to X is 53 inches
long and subtends an angle of 47 13' at Y, find the length of
the chord YP.
Note. Use cosecants in questions 17 to 20.
17. How long must a ladder be which just reaches a window
15 feet above the ground, if it makes an angle of 53 19' with
the ground 1
18. What is the diameter of a circle in which a chord
5 inches long subtends an angle of 25 28' at the circumference ?
[Draw the diameter through one end of the chord.]
19. If the slope of a road is 4 6', what is its gradient?
(Of. Ex. 20, Chap. III.)
20. If the line joining the tops of two vertical posts makes
an angle of 17 13' with the horizontal, and if one post is 6 feet
longer than the other, what is the distance between the tops 1
EXERCISES ON THE Six RATIOS
21. Draw a freehand sketch of a triangle ABC having C = 90,
o = 4 inches, b = 3 inches. From this figure find the six trigono-
metrical ratios of A and of B.
Find the size of the angle A by looking up sin A in the Table
of Natural Sines, and verify the remaining five ratios by searching
for this angle in the appropriate Tables. Repeat for the angle B
and arrange all results in columns.
22. Repeat question 21 for a triangle having C = 90,
c M 40 cm. and 6=9 cm.
23. Observing that the angles A and B in questions 21 and
22 are complementary, what do you note with regard to the
ratios of complementary angles)
24. If tan A = T 6 r , find (by drawing a freehand sketch) the
other five ratios of A and write them down. Verify from the
Tables as before.
25. If sin as= ^j., find the other ratios and verify.
V] AND COTANGENT 23
26. Find the sine, cosine and tangent of 35 from the
Tables. Calculate the value of ^- , and of sin 2 35 + cos 3 35.
cos 35
[sin a 35 means the square of the sine of 35.]
What do you observe with regard to these values!
27. Repeat question 26 with any angle chosen at random.
Do the same relations hold good ?
28. Prove that these relations are true for any acute angle
by drawing a sketch of a right-angled triangle and making use
of Pythagoras' Theorem.
29. Making use of your tables if necessary, draw a rapid
freehand sketch of the graph of sin A when A has values between
and 90.
30. As A increases from 80 and becomes more and more
nearly equal to 90, to what value does PN (see figure of Ex. 6,
Chap. III.) more and more nearly approach ?
What then is the ultimate value of sin 90 ?
Write down the values of cos 90, tan 90% sin 0% cos 0,
tan 0.
31. Draw rapid freehand sketches of the graphs of cos A
and tan A for values of A between and 90.
32. What can be asserted with regard to the greatest and
least values of the sine of an acute angle ?
What can be asserted with regard to the cosine and tangent
of an acute angle 1
33. By means of the Tables find a relation between tan* 35
and sec* 35.
34. Choose an angle at random and show that the relation
is still true.
35. Prove as before (question 28) that this relation is true
for any acute angle.
36. Find and prove a relation between cosec'A and cot 9 A
when. A is an acute angle.
CHAPTER VI
MISCELLANEOUS PROBLEMS INVOLVING RIGHT-ANGLED
TRIANGLES
Note. At this stage it is most essential that the pupil should be able to
recognize immediately the ratios of the angles in a right-angled triangle and
readily to express the length of one line in terms of another. Constant
practice with figures similar to those in questions 1 and 6 below is
recommended for pupils who hesitate.
Revise Ex. 6, Chap. I.
1. In the figure C = 90 and p is perpendicular to c cutting
it into two parts x and y.
C
Fig. 17.
Which other angle in the figure is equal to B, and why?
Show that the whole triangle and the two triangles in to 'which p
divides it are all equiangular (and therefore the same shape).
Now - sin B. Write down two other fractions equal to sin B.
a
CH. VI]
MISCELLANEOUS PROBLEMS
2. Write down three different expressions for cos B, tan B,
sec B, cosec B, cot B, in the figure of question 1.
3. Write down three expressions for each trigonometrical
ratio of the angle A in the figure of question 1.
4. If in the figure of question 1, a = 20 cm. and 2? = 12 cm.,
find B, A, 6, x, y and c, and verify the results of questions 1 3.
5. If in the figure of question 1,6 = 6-5 cm. and p = 6 cm., find
A, B, a, x y y and c.
T i ,. AN
6. In the figure = cos a,
.*. AN = x cos a.
90'
Pig. 18.
Write down in a similar way the values of p, y, AC, CN, in
terms of x and a.
7. In the figure of question 1 what lengths are represented
by a cos B, b sin A, c cos B, a cos A, p tan B, p cot B, p sec A,
p cosec A, b sin B ?
8. In the figure of question 6 what lengths are represented
by x sin a, x tan a, p cot a, y cos a, p tan a ? '
26 MISCELLANEOUS PROBLEMS [CH.
Without using Pythagoras' Theorem solve the triangles in
which C = 90 and
9. B = 2419', a =10-4.
10. A = 29 53', a =20-1.
11. A = 44 31', c = 2-4.
12. a = 24-7, 6=14-6.
13. c = 140, a = 77.
14. A = 72 13', 6=15-5.
15. B = 3t37' 1 c=1200.
16. 6 = 5-7, c=ll-4.
17. c=!G-75, a = 3-35.
18. c=3a.
Note. In the following questions the answers may be obtained in a
variety of ways, but by using the appropriate ratio the answer may be
arrived at by one operation.
19. From the top of a cliff 250 feet high the angle of
depression of a boat is found to be 17 43'. How far is the boat
from the top of the cliff?
20. A ship sails 17 miles due N. and then 25 miles due E.
In what direction does the starting-point lie ?
21. The angle of elevation of the top of a tree is found to be
29 11'. If the observer is 120 feet from its base, how high is
the tree ?
22. The string of a kite (supposed to be straight) is 225 feet
long and is inclined to the ground at an angle of 52 5'. Find
the height of the kito above the ground.
23. What is the slope of a railway line whose gradient is
Iin60?
24. Find the length of one of the equal sides ot an isosceles
triangle whose base is 12-4 cm. long and whose base angles are
each 41 28'.
Vl] INVOLVING RIGHT-ANGLED TRIANGLES 27
25. Find the radius of a circle in which a chord 3-4 cm. long
subtends an angle of 104 at the centre.
26. From the top of a flagstaff 108 feet high a wire 125 feefc
long is stretched to the ground. What is the angle between the
wire and the flagstaff 1
27. In a triangle PQR, PQ = 1 2'2 cm. and Q = 39 '37'. What
is the length of the straight line PN drawn from P perpendicular
toQR?
28. Find also the length of QN in the last question.
29. If in the question above, QR = 17'8 cm., what is the
length of NR? Hence find the angle at R.
30. An aeroplane is observed from a point A and its angle
of elevation is found to be 67 32'. If at the same instant it is
exactly above B (on the same level as A), and AB = 75 yards, how
high is it ifbove B 1
31. The angle A of a parallelogram ABCO is 31* 3' and the
perpendicular distance between AD and BC is 3*52 inches. Find
the length of the side AB.
32. PN is a tangent to a circle whose centre is O, N being
on the circumference. If the radius is 5-7 cm., and OP = 9*5 cm.,
what angle does PN subtend at the centre?
33. If a door 4 ft. 2 inches wide is opened through an angle
of 112 Q 42', how far (in a straight line) is the bottom corner from
its position when the door was closed ]
34. What is the radius of a circle which touches the five
sides of a regular pentagon if each side is 5 cm. long ? [Find the
angle at the centre,]
35. Find the radius of the circle which passes through the
five angular points of a regular pentagon whose sides are each
5 cm. long.
36. Repeat questions 34 and 35 for a regular hexagon
(6 sides).
28 MISCELLANEOUS PROBLEMS [CH.
37. Repeat for a regular decagon (10 sides).
38. A ship is sailing N. 11 15' E. at 9 miles an hour. At
what rate is she moving due N. ?
39. From a point 27 feet from the base of a pole and on the
same level, the angle of elevation of the top is 63 26'. Calculate
its height to the nearest inch. Find also the distance of the
point of observation from the top.
40. Find the angle of elevation of the top of the pole in
the last question from a point 20 feet from the base.
41. A rod 5 inches long is hinged at a point on the ground,
and from the other end hangs a plumb-line. What angle does
the rod make with the plumb-line when the top of the rod is
(a) 4 inches above the ground, (b) 2 inches above the ground 1
42. A pole is 126 feet high. A man 6 feet high stands at
a distance of 100 feet from the foot of the pole and observes the
angle of elevation of the top of the pole. What does he find the
angle of elevation to be I
43. A diameter AB of a circle is 5 cm. long. P is a point on
the circumference 2-34 cm. from A. Find the angle subtended by
PA at B.
44. The shadow of a vertical stick 4 feet high is found to be
5*72 feet long. Find the altitude of the sun.
45. A lies 5 miles due North of B, and C lies 6 miles due East
of B. Calculate the distance and bearing of C from A.
46. At a certain Marconi Station a vertical pole 96 feet
high is made in three equal sections and supported by wire stays
stretched from the top of each section and fixed to points in the
ground. If wires from the top of each section are fixed to a
point on the ground 25 feet from the base of the pole, Calculate
the angles which they make with the ground.
47. Find the height of an isosceles triangle whose base is
10 cm. long and whose- vertical angle is 35 28'.
VI] INVOLVING RIGHT-ANGLED TRIANGLES U9
48. In the figure, CP represents the connecting- rod and OP
the crank of a steam engine. P revolves round O, and C moves
backwards and forwards along OC.
If CP is 40 inches long and OP is 12 inches long, find the
angle at C when OP is at right angles (1) to CP and (2) to OC.
49. ABC is a triangle right-angled at C. If BC = 17*3 feet
and if ABC = 30 27', calculate the length of AB to the nearest
inch.
50. From the top of a cliff 300 feet high the angle of
depression of a ship is 13 46' and the angle of depression of a
rock between the ship and the shore is 32 10'. Calculate the
distance of the ship from the rock (to the nearest foot).
51. On one occasion while the Atlantic cable was under
repair, it was found that when the grappling iron seized the cable
on the ocean bed, 4890 fathoms of hawser had been paid out, and
that when drawn taut the hawser made an angle of 72 with the
horizontal. What was the depth of the water at this spot in
fathoms ?
52. How many degrees are there in an angle in a semicircle 1
In a semicircle of diameter 10 inches, a chord 4 inches long is
put, one, end of which coincides with one end of the diameter.
Calculate (a) the angle between the chord and the diameter, and
(b) the length of the chord joining the other ends of the diameter
and chord.
30 MISCELLANEOUS PROBLEMS [CH.
53. A diameter AB of a circle bisects a chord CD at right
angles at O. The angle OCA = 50. Find the lengths of AO
and BO, given that the chord CD is 20 inches long. What are
lengths of AC and BC ]
54. A lighthouse bears 5 miles due N. of a cruiser steaming
N. 18 -6 E. By how many miles does the cruiser clear the
lighthouse ?
55. A cruiser and a torpedo boat set out at the same time
from point C. The cruiser steams at the rate of 20 knots in
a direction N. 15 24' E., and the torpedo boat at the rate of
30 knots in a direction N. 74 36' W. Calculate the distance
and bearing of the torpedo boat from the cruiser at the end of
30 minutes.
56. A, B, and C are buoys. The beciring of B from A is
N. 47 R, and the bearing of C from A is S. 43 E. B is known
to be 5 miles from A, whilst C is due S. of B. Find the distances
of C from B and A.
57. A flagstaff on the far bank of a river is seen by a man
immediately opposite to subtend an angle of 57, whilst on
retiring 100 ft. the elevation is only 35. Find the breadth
of the river to the nearest foot.
58. The Rock of Gibraltar is 1396' in height; the angle of
elevation of its summit from a ship is 22: how far will the ship
have to move directly towards it before its angle of elevation
is 31?
59. A straight railroad rises 2-4 feet in every 100 yards of
rail : find its inclination to the horizontal.
60. A straight line from Southampton pier to Ryde pier
subtends 90 at Portsmouth pier; Southampton pier lies
15*2 miles N. 46 W. from Ryde pier, and Portsmouth pier
bears N. 28 E. from Ryde pier ; find the distance between
Portsmouth and Rvde Diers.
VI] INVOLVING RIGHT-ANGLED TRIANGLES 81
61. At Southampton on Midsummer's Day the altitude of
the sun is 62 45'. If a mast is 100 feet high, find the length of
its shadow.
Find also the length of its shadow on December 21st, when
the altitude of the sun is 15 45'.
62. From the top of a vertical cliff 200' high the angle of
depression of a boat anchored out in the bay is 4G 28 ; . Find
how far the boat is from the foot of the cliff.
63. A kite is held by a string 100 yards long, and its angle
of elevation is found to be 50 16'. What is the vertical distance
of the kite from the ground? (Neglect the sag of the string.)
64. A ship X is 10 miles S. 52 W. of a harbour at the
moment that another ship Y is leaving the harbour. If Y
steams S. 38 E. at 8 knots and the ships meet in 2 hours,
find X's course.
65. From the top of a tower A, 60 feet high, the angle of
depression of the foot of another tower B is 23 16'; whilst from
the foot of A the elevation of the top of B is 45 17'. Find how
far A is from B, and also the height of B to the nearest foot.
66. A ship on leaving port steams 10 miles N. 50 11' E.,
and then 5 miles N. 39 49' W. Find the distance she must go,
and the course she must steer, to reach port again.
67. In a circle of radius 27 inches find the length of a chord
which subtends an angle of 163 10' at the centre. Find also
how far this chord is distant from the centre of the circle.
68. From a ship sailing on a course due N. a lighthouse,
13 miles away, is observed to bear N. 19 34' E. When the
ship is nearest to the lighthouse, how far from it will she bel
What will be the bearing of the lighthouse from the ship
when they are 7 miles apart t
32 MISCELLANEOUS PROBLEMS [CH.
69. (The diagonals of a rhombus bisect each other at right
angles and also bisect the angles through which they pass.)
Make use of these properties to find the angle at B of the
rhombus ABCD, the length of each side of which is 10 inches,
the length of the diagonal AC being 5 '234 inches.
70. Observations were taken from a point 500 feet from the
foot of a tower with a flagstaff on the top, and the angles of
elevation of the top of the tower and of the top of the flagstaff
were found to be 28 15' and 30 47' respectively. Find (1) the
height of the tower and flagstaff; (2) the height of the tower
alone ; and hence (3) the height of the flagstaff alone.
71. A ship sails 4 miles due West and then 3 miles due
North ; find its distance from the starting point and its bearing
from that point.
72. A bears 10 miles due N. of C, and B bears N. 25 E.
of C, and B bears S. 65 E. of A. Find the distances of B from
A and C.
73. A lighthouse 10 miles away is seen to be 15 off a ship's
course. At what distance will the ship pass the lighthouse if
she holds her course?
74. At 10 a.m. a ship is observed from a lighthouse to bear
9 miles N. 57 32' E. and is known to be sailing S. 32 28' E.
At 11 a,m. her bearing is S. 6045'E. Find (1) rate of ship's
sailing ; (2) distance of lighthouse at second observation.
75. In a triangle ABC, the angle B 36 52', AB = 4-5 inches,
and BC = 4-68 inches.
By drawing AD perpendicular to BC, and calculating the
length of AD and BD, find the length of DC and AC, and the size
of the angle C.
Make a table of the sides and angles of the triangle ABC.
VI] INVOLVING RIGHT-ANGLED TRIANGLES 33
76. A theodolite was used to determine the height of a flag-
staff and the following data were obtained :
Height of theodolite telescope above ground 3 ft. 4 in.
Distance of theodolite from foot of flagstaff 120 ft
Angle of elevation of the top of the flagstaff 27*5.
Calculate the height of the flagstaff.
77. A and B are two points on opposite sides of a tower.
The distance AB is 140 feet. The angles of elevation of the top
of the tower from A and B are respectively 62 and 31. Cal-
culate the height of the tower.
78. O and P are points on a straight stretch of shore 1 mile
apart, and O bears N. 74 W. of P. From a ship at sea O bears
N. 15 W. and from the same ship P bears N. 75 E. Calculate
the distance of the ship from O and also its distance from the
nearest point of the shore.
79. The ropes of a swing are 25 feet long and the height of
the seat above the ground at the highest and lowest points are
14 feet and 3 feet respectively.
What is the angle through which the swing moves from side
to side 1
80. A flagstaff 90 feet high subtends an angle of 45 at a
point A on the ground due South of it. Find the angle of eleva-
tion of the top of the flagstaff at a point B, 120 feet due East
of A.
81. The Tay Bridge extends North and South a distance of
1*764 miles. A man walks due East from the North end of the
bridge until the bridge subtends an angle of 33 17' at his eye.
How far is he now in a straight line from the South end of the
bridge ?
82. A flagstaff consists of two poles one fixed above the
other. From a point 64 feet from the base the angles of elevation
of the tops of these poles are found to be 40 29' and 61 51'.
Find the length of each pole.
34 MISCELLANEOUS PROBLEMS [CH. VI
83. From the top of a cliff 320 feet above sea level the
euigles of depression of two boats in a line are found to be 11 21'
and 34* 19'. Find their distance apart.
84. From a ship sailing due S. at 12 miles an hour the
directions of two objects on shore are observed to be S. 29 14' W.
and S. 43 7' W. After sailing for 25 minutes they are observed
to be in a line due W. of the ship. Wfcat is the distance between
them and how far was the ship from each of them at the first
observation ?
85. (See figure of Ex. 56, Chap. II.) If A is the vertex of a
square pyramid whose base is BCD, and if BC = 10 cm. and
AB= 12 cm., find (1) the angle between AB and BC, (2) the angle
between AB and BD, (3) the inclination AKN of a side-face.
86. Find the height of a square pyramid each of whose edges
is 5 inches long. Find also the inclination of (1) a side-edge,
(2) a side-face to the base.
87. If the edges of a rectangular block are 2, 3 and 6 inches
respectively, find the angle between a diagonal of the block and
the diagonal of each face.
88. The angle of elevation of a tower 250 feet high is
observed from A to be 20. How far is A from its foot? If A is
due E. of the tower and due N. of a point B from which the
tower bears N. 40* W., how far is B from the foot? Find the
elevation of the top from B.
89. Repeat question 88 making height of tower a? feet and
18 the angle of elevation from A.
90. P is 1000 yards due N. of Q and on the same level. An
airship is observed from P and Q simultaneously. From P its
elevation is 23 and it bears due W. From Q it bears N. 42 W.
Find its height above the ground.
CHAPTER VII
AREAS
1. What is the area of a rectangle whose adjacent sides are
(1) 5 cm. and 10 cm., (2) 131*4 yards and 101'2 yards?
2. Draw a sketch of each of the rectangles above and draw
a diagonal.
By considering these figures find the area of a right-angled
triangle whose shorter sides are (1) 5 cm. and 10 cm., (2) 131*4
yards and 101*2 yards.
3. What is the area of each of the right-angled triangles
ANB, ANC in the figure? What is then the area of the whole
triangle ABC?
A
4cm
N
Fig. 20.
4. How is the area of a triangle found when the base BC
and the height AN are known 1 Express the answer quite clearly
in words.
86 AREAS [CH.
5. If two sides of a parallelogram are 7 cm. and 5 cm. long
respectively and one of its angles is 40, what are the perpen-
dicular distances between the opposite sides ? The parallelogram
is twice the triangle ABC. Calculate its area, (See Fig. 21.)
Fig. 21.
6. How is the area of a parallelogram found when the sides
and the perpendicular distances between them are known 1
7. How is the area of a parallelogram found when two sides
(as, y) and the included angle (A) are known ?
8. How is the area of a triangle found when two sides and
the included angle are given ?
9. Find the area of each of the following triangles :
(1) a = 5 cm., 6 = 4 cm., C = 30.
(2) 6 = 10 cm., c = 3 cm., A = 36 52'.
(3) 6-6-49 cm., c = 5*73 cm., A -65 29'.
(4) a = 21-43cm., c = 13-91cm., B = 5219'.
10. In a circle of radius 10 cm., a chord AB subtends an
angle of 35 at the centre O. Calculate the area of the triangle
OAB.
11. What is the area of a regular hexagon (six
inscribed in a circle of radius 10 cm. ?
(Find angle subtended by one side at the centre.)
12. Find the area of regular figures of 5, 7, 8, 9 sides
inscribed in a circle of radius 10 cm.
VII] AREAS 37
13. Find the area of regular figures of 5, 6, 9 sides described
with their sides touching a circle of radius 10 cm.
14. The area of a rhombus (parallelogram with equal sides)
is 648 square yards and one of its angles is 150*. Find the
length of one side.
15. The area of a rhombus is 14'58 sq. cm. and one diagonal
is twice as long as the other. Find its angles and sides.
16. The base of an isosceles triangle is 4 cm. long and the
area is 20 sq. cm. Find all the angles of the triangle.
17. Find the area of each face and the area of the whole
surface of each of the solids given in Exs. 56, 57, Chap. IL
18. A regular hexagon (6 sides) is inscribed in a circle oi
radius 7 cm. What is the area of the circle ? If two adjacent
corners A and B be joined to the centre O, what is the area of th<
sector AOB 1 Find also the area of the triangle AOB and hence
the area of the segment cut off by AB.
19. Find the area of the segment cut off by one side of a
regular pentagon (5 sides) inscribed in a circle of radius 10 cm.
20. If two tangents OA, OB be drawn to a circle of radius
4 cm., and if L AOB = 30, what is the area of the figure bounded
by the tangents and the larger arc AB
CHAPTER VIII
LOGARITHMS
L 10 3 means the product of three tens or 10 x lOx 10.
What does 10 4 mean? How many tens must be multiplied
together to be equal to 10 3 x 10M
Express 10 8 x 10 4 as a power of ten.
What is done to the 3 and 4 to obtain the new index 9
Express 10 a x 10 7 and 10 8 x 10 9 each as a power of ten
2. To assign a meaning to 10' 5 (or
[" The product of half a ten " has no intelligible meaning.]
What was done with the given indices to obtain the new
index in each of the sums in question 1 ]
Use the same rule to express 10^ x 10^ as a power of ten.
Now 4x4 = 16, and 4 is called the square root of 16.
What is then the meaning arid value of 10* (or 10 08 )?
Remember this result.
3. Express 10* x 10* x 10* as a power of ten. What is the
meaning of 10M (Compare with question 2.)
What is the meaning and value of 10* (or 10 " 25 )!
4. Express 10* x 10* as a power of ten and find its value.
SH. VIII] LOGAEITHMS 39
5. Make a table showing the values of 10 ' 28 , 10 10 075 , 10 1<0 ,
and use it to draw a graph on squared paper showing the powers
of ten which are equivalent to numbers between 1 and 10.
From this graph find the numbers which correspond to 10'*,
10' 4 , 10 08 , 10 ' 88 . Also make a table showing the powers of ten
which correspond to the numbers 2, 3, 4, 5, 6, 7, 8, 9; 10.
The index of the power of ten which is equal to 3 is called
the logarithm of 3 (to base 10). That is, log 3 =0-4771, for
3 = 10o-477i. and log 10000 = 4, for 10000 = 10 4 . This should be
remembered.
6. Turn to the printed Table of Logarithms and search for
the logarithms of the whole numbers from 2 to 10.
[The position of the decimal point is not shown in the table.]
Add a column to the table constructed in question 5, and
compare the values obtained from the graph with the values
given in the printed Tables.
7. The arrangement of the Table of Logarithms is very
similar to that of Tables of Natural Tangents, etc., already
familiar.
Now 3-5 = 10' c441 , 3-56 = 10 ' 8514 , 3-563 = 10' M18 . Search for
these in the printed Tables. Then write down each of the
following and express it as a power of ten by means of the
printed Tables :
3-7, 3-75, 3-758, 7-64, 7-649, 9-8, 9-805.
8. Find from the Tables the numbers which are equal to
100-3010 j 00'8808 JQO'4398 JQO f 4409 JQO'4404
100789C 1007903 1007901 J 00-8819 100'81.
(These may be found in the Tables of Logarithms or in those o:
Antilogarithnis.)
40 LOGARITHMS [OH.
9. From the Tables 3-563 = 10 08518 and 1 -212 = 10 ' 08 ".
Therefore 3-563 x 1-212 10 ' 8518 x 10 ' 0888 .
Now 10* x 10 8 = 10 8 by adding the indices ; thus by the same
process
3-563 x 1-212 = 10 ' 6518 x lO ' 0886 ^ 10' 63M .
Write down this line of work and find from the Tables the
number whose logarithm is 0*6353 ; this is the product of
3-563 x 1-212.
Verify by multiplying these two numbers in the ordinary way,
but remember that values given in the Tables are not quite exact ;
they are correct only to four significant figures.
10. Find the value of each of the following by means of
logarithms and verify the answer in the usual way :
(1) 2x1-427. (2) 3x2-379. (3) 2-5x3-624.
(4) 4x1-767. (5) 3*5x2-147.
11. Find by using logarithms the value of each of the
following and verify by ordinary multiplication :
(1) 4-651 x 1-843. (2) 3*794 x 2-156.
(3) 1-732x1-732. (4) (2-54) 2 .
(5) 7-342x1-112. (6) (3-152)*.
10
12. It is clear that y^ 8 = 10 a by subtracting the indices.
T .1 3 ' 563 1AO-4683
In the same way = 10 04683 .
Write down this line of work and find from the Tables the
number whose logarithm is 0*4683. This number is the value
of ,777, . Verify by long division.
*
VIIl] LOGARITHMS 41
13. Find by logarithms the value of each of the following
and verify by division :
1X 4-794 /ov 6-315 /ox 9-637
W (2) ' (3) "2^5- '
MN 3-295 7-694 5-407
< 4) T3T- < 5) ?795- < 6 > R36*
LOGARITHMS OF NUMBERS GREATER THAN TEN
14. Hitherto only numbers between 1 and 10 have been dealt
with. Now consider a number greater than 10, for instance 356*3,
356-3 = 100 x 3-563 = 10 2 x 10 M18 = 10 aw
by adding indices as before.
Write down 35-63, 3563, 35630, 356300, and express each as
a power of ten in a similar way.
15. In question 14 what do you observe with regard to the
part of each logarithm after the decimal point ?
(This part is sometimes called the mantissa, which in Latin means " a
small part added." The whole number before the decimal point is called the
characteristic.)
Explain in words why it is that no decimal points are shown
in the Table of Logarithms.
16. Write down each of the following and express it as a
power of ten. (It is advisable to express each as in the first
example of question 14.)
463-4, 79-18, 16-29, 721*4, 72-14,
3114, 76300, 204600, 73-51, 213'5.
17. Find the numbers which are equal to 10 21J51S , 10 1W8B ,
79 , 10 14300 .
LOGARITHMS
[CH.
18. Find the value of each of the following by logarithms :
(1)
2-572x14-63.
(2)
31-79 x 43-72.
(3)
727-6x1-571.
(4)
6297-x 23-67.
(5)
97-64x94-67.
<)
212-5x409-5.
(7)
21-35x37-67.
(*)
3-244 x 315-4.
(9)
61-12x43150.
(10)
73050 x 4350.
(11)
14-63
(12)
43-72
2-572 '
31-79*
(13)
727-6
lETi'
(14)
6297
23-67*
(15)
57-93
41-52*
(16)
7296
'47 -
(17)
312400
76-53 '
(18)
49-35
17-21*
(19)
13-69
(20)
200-5
"OHf"
13-61 '
LOGARITHMS OF NUMBERS LESS THAN UNITY
19. All the numbers treated above are greater than 1. In
order to find the logarithm of a number less than 1 (say 0*03563),
it is necessary to assign a meaning to expressions such as 10~ 8
(we know that 10 3 means "the product of three tens" but "the
product of minus three tens " is not intelligible).
Write down the following in a column and insert their values :
10 J = 100,000
10 4 =
10* =
10* =
Note that the indices on the left-hand side decrease by one
each time. How is each of the values on the right-hand side
obtained from the one above it? Continue the column downwards,
VlllJ LOGARITHMS 4&
decreasing the indices as before and find .values of 10, 10" 1 , 10~ 8 ,
10-, 10- 4 , 10~ 5 . From this it will be found that 10~ 8 = ^J^-T. or
~jr- 3 , and this has an intelligible meaning. Remember this result.
20. Now to find the logarithm of 0*03563 we have
0-03563 = ^j^ = JTJQ * 3*563 = 10- 2 x lO^^lO- 8 * 18 by adding
indices as before, and this is written 10* M18 to show that the
minus sign applies to the 2 only. This arrangement is adopted
in order to simplify the Tables, as will be seen hereafter.
By writing each of the following in the same way as in
question 20 above, express it as a power of ten : 0*01212, 0*1212,
0-3563, 0*0003563.
21. Write down the following in a column and express each
as a power of ten : 3563, 356*3, 35-63, 3*563, 0-3563, 0-03563,
0-003563, 0*0003563. By adopting the system of writing the
minus sign over the characteristic (e.g. 2*5518), it is found that
the same Table of Logarithms may be used for numbers greater
than or less than 1.
Discover and express in words a rule connecting the
characteristic of the logarithm with the position of the decimal
point of the number it represents.
22. Express each of the following as a power of ten :
0*0345, 0-7691, 0*00463, 0*0001105,
0*9971, 0-9065, 0-0435, 0-5432.
Note. To find the value of 0-02 x 0-007 we have
0'02 x 0-007 = 10 5>301 x 10 2-3010
3-8451
=0-00014. 4-1461
The addition is done in the usual way, but it must be remembered that
2 and 3 are simply - 2 and - 3.
44 LOGARITHMS [OH.
23. Find the value of each or wie following products by
logarithms and verify each one :
(1) 0'5x40. (2) 0-004x50. (3) 0-025x160.
(4) 0-5x0-5. (5) (0-04) 2 . (6) 0'04 x 0-05.
(7) 0-012 x 0-2. (8) 0-0008 x 40. (9) 0-0011 x 0-02.
(10) 0-25 x 0-0003.
24. Find the value of the following by logarithms :
(1) 0-02572x0-1463. (2) 0-02367x0-6297.
(3) 0-7276x0-01571. (4) 0-4372x0-003179.
(5) 946-7 x 0-09764. (6) 3-244 x 0-3154.
Note. To find the value of 0-2 H- 0-007 we have
0-2 IQI'3010
0-0007 10^ 51
The subtraction is best effected by dealing with the 3 first n
The operation 1-3010
8-8451
1-4559
is then the same as 2-3010
0-8451
1-4559
for - 3 is the same as + 3. Check by adding the last two numbers.
25. Find the value of each of the following by logarithms :
0-001463 m 0-7276 0-002367
W ^2572 ' () 0-01571' () 0-6297
/ ' 43 H ) ^ 764 (Q) 3 ' 244
W 6-003179 ' w 0-9467' ^ ; (F3T54
VIIlJ LOGARITHMS 45
26. In questions 2 and 3 of this Chapter it was found that
10* meant VlO and 10* meant ^To.
In the same way 1/50 = (50)* = (10 r8900 )4.
Now (10 4 ) s =10 4 x 10 4 xl0 4 =10 12 , the new index being ob-
tained by multiplying 4 by 3.
In the same way (lO 1 * 990 )* = lO ' 6863 by multiplying 1-6990 by
J, i.e. by dividing it by 3.
Find the number whose logarithm is 0-5663. It is <s/50.
27. In a similar manner find the values of
N/320,
28. In dealing with the logarithm of a number less than 1,
remember that the decimal part (or mantissa) must always be
kept positive, the characteristic being negative.
What is half of 3-3010 (i.e. - 3 + 0-3010) ?
To transform the answer so that the decimal part remains
positive, it is most convenient to think of 3*3010 as being equal
to 4+1-3010. Dividing this by 2, the logarithm of N/-002
may be found, and hence its value from the Tables.
29. In a similar manner tind by logarithms the value of the
following, verifying the answers :
J&QOOi.
(1) -s/004.
(4) ^O-OOS.
(2) 76 : 004.
(3)
(6)
(9)
(7) Vo-765.
{ioy -yo-oofs.
(8)> ^0-057.
LOGARITHMS
[OH.
MISCELLANEOUS EXAMPLES ON LOGARITHMS.
When the principles have been grasped it is convenient to arrange the
work as in the following Examples.
No.
A. 23-57x3-721=87-70.
372-1
2^357
= 157-9.
Find the following :
(1) 72-47 x 26-05.
23-57
3-721
log
1-37124
0-5706
No.
372-1
2-357
1-9430
log
2-5706
0-3724
2 -1982
(2) 631-4 x 0-05641.
,,s 326-4
w
3-461 *
w
435-6*
(5)
(17-49)'.
( 6 )
(31-62) 8 .
(7)
>/29 T 95.
(8)
#12.
(9)
(0-7462) 4 .
(10)
(0-06925) 8 .
(11)
N/0-007534.
(12)
#0-04691.
(13)
s/(0-1193)'.
(14)
#(0-007 2) 2 .
0-3157
/73-91V
(Ib)
(16)
0-04694 *
(17)
'/ 0-0724
(18)
7-529x0-04392
V 7-4 x 3-142'
0-5497
no\
3-142
/9m
(0-325) 8
0-247 x(0-04627)*
VIII]
LOGARITHMS
47
Note. In working out an example such as ^J5 it is convenient to
14o
make a skeleton arrangement before looking out logs from the Tables, thus :
57
sin 69
148
The logarithm of sin 69 may be found direct from the Table of Log-
arithmic Sines ; in some Tables 10 is added to the characteristic to avoid
the clumsy printing of minus signs above the numbers. The work then
stands thus :
67 x sin 69
143
=0-3722.
Find the value of :
(21) 16 sin 35*.
(23)
(25)
(27) 1341 x 7-eixcos 14* 19'.
(29) 73-56 coa 19 21'
61 sin 49
57
sin 69
1-7559
1-9702
143
1-7261
2*1553
1-5708
11-61 tan 49.
114 tan 19 24'.
243 cos 47.
63 sin 17 24'.
1 -64 cos 23 47'.
(22)
(24)
(26)
(28) gj
6 1-42 sin 25 52'.
13 sin 29 46'
(31)
(32)
(33)
(34J
(35)
95
loft sin 43 17'
12 - 6x sin^F24"
2 x 3-24 x 5-62 cos 72 31'.
2 x 7-63 x 8-5 x cos 31 37'.
? sin 23' 14'. (36)
14
19-43 sin 73 29'
10-65
CHAPTER IX
SOLUTION OF TRIANGLES WHICH ARE NOT
RIGHT-ANGLED
Note. The calculations in this and the following chapters may often be
much simplified by the use of logarithms.
A. Given two sides and the angle included.
1. Draw accurately a triangle ABC having angle A = 60,
6=10 cm. and c = 9 cm. Measure B, C and a.
2. If, in the triangle above, CN is drawn perpendicular to
AB, the right-angled triangle ACN can be solved by calculation,
and then the parts of the triangle BCN can also be calculated.
Calculate B, C, a by this means.
3. Solve the same triangle by drawing a perpendicular from
B to AC.
4. Draw a freehand sketch of the triangle ABC in which
A = 53 8', b = 7-5 cm., c = 9cm. Solve it by calculation and
verify by accurate drawing.
5. Repeat for a triangle having A = 23 35', 6 = 10 cm. and
c- 12 cm.
6. In the questions 1, 2, 3 above it was found possible to
solve the triangle by drawing a perpendicular from C or from B.
Is it possible to do so by drawing a perpendicular from A?
Express very carefully in words the line (or lines) which should
be drawn in order to solve a triangle in which two sides and the
included angle are given.
OH. IX] SOLUTION OF TRIANGLES 49
7. Solve the following triangles and verify the answers by
drawing to scale.
Note. An example of orderly arrangement will be found in the Appendix.
(1) a = 25, 6 = 24, C = 4125'.
(2) a =2-6, c=2-3, B = 6124'.
(3) 6 = 44-7, c = 46-9, A = 24 17'.
8. If A = 60, 6 = 10 cm., c = 3 cm., solve by drawing a per-
pendicular from B to AC.
9. Solve the triangle of question 8, by drawing CN perpen-
dicular to AB produced and dealing with the right-angled triangles
ACN and BCN.
10. Solve the triangle in which A = 126 52', 6 = 10 cm.,
o = 7*5 cm. by drawing the perpendicular from B to AC produced.
11. Is it possible to solve the last triangle by drawing a
perpendicular (1) from A, (2) from B]
If either of these is possible, complete the solution.
12. Solve the following triangles by drawing a perpendicular :
(1) 6 = 23-4, c = ll-6, A = 37 49',
(2) a =149, 6 = 51, C = 5823'.
(3) a = 26-4, b = 31-7, C = 135 16'.
(4) 6 = 3-49, c = 4-37, A = 98 23'.
(5) c = 0'45, a = 0-19, B = 16147'.
B. Given two angles and one side.
1. Draw as accurately as possible a triangle ABC in which
A = 53 8', B = 59' 29' and c = 10 cm. Measure C, a and 6.
2. From B drop a perpendicular BN to AC and calculate
C, AN, NB, NC, CB and CA in this order.
3. Draw a sketch of the triangle given above and write
down how it can be solved if a perpendicular is drawn from A
instead of B. Indicate the parts to be found in their proper
order as in question 2.
4
50 SOLUTION OF TRIANGLES WHICH [CH.
4. Is it possible to solve the triangle by drawing a perpen-
dicular from C? Describe in words the line (or lines) which
should be drawn to solve a triangle when two angles and the side
between them are given.
5. If two angles and the side opposite to one of them (A, B, a)
be given, from which angle should a perpendicular be drawn in
order to solve the triangle 1
Is it possible to solve it in more than one way? Indicate
how the triangle should be solved.
6. Solve by this method the triangles in which
(1)
A = 43 27',
B = 619',
a = 3-5.
(2)
A = 71 42',
B = 6519',
c = 16-9.
(3)
A = 29 17',
B = 108 53',
6 = 214.
(4)
A = 13 39',
= 41 11',
a =14-63.
(B)
= 112 41',
A = 47 25',
6=29-46.
C Given three aides.
1. Draw a sketch of the triangle ABC in which a = 5 cm.,
6 = 6 cm. and c = 4cm. Draw AN perpendicular to BC. Let
BN = a; cm. What is then the length of CN ?
Using Pythagoras' Theorem in the triangles ABN, ACN, find
two expressions for the square on AN and by equating them find
the length x. Complete the solution of the triangle ABC and
verify by drawing an accurate figure.
2. Verify the results of question 1, by drawing the perpen-
dicular from C and solving as before.
3. Draw the perpendicular from B in the same triangle and
indicate (without actual calculation but by writing down the
parts to be found in their proper order) how the triangle may be
solved.
IX] ARE NOT RIGHT-ANGLED 51
4. Solve the following triangles. Consider carefully which,
perpendicular will lead to the simplest calculation:
(1)
a =
7
cm.,
b
=
8cm.,
c
= 10cm.
(2)
a =
4,
b
=
3,
c
9
(3)
a =
2
5,
b
=
3,
c
= 1.
(4)
a =
2
5,
b
=
3-5,
c
= 4.
(5)
a =
6
34,
b
=s
7-42,
c
= 3
5.
D, Given two sides and an angle opposite one of them,
1. Draw as accurately as possible a triangle having
A = 36 52', 6 = 10 cm., a = 8 cm.
Measure B, C and c ; there are two answers to each.
2. Draw CN perpendicular to AB in the triangle above and
calculate the values of B, C and c by solving the triangles ACN
and BCN.
3. Is it possible to solve the same triangle by drawing the
perpendicular (1) from B, or (2) from A? Illustrate the answer
with a sketch.
4. Are there two possible values for c in a triangle having
A = 36 52', b = 10 cm. and a = 12 cm. 1
Solve this triangle completely.
5. In the triangle of question 1, a is less than 6, and in
question 4, a is greater than 6. Express clearly in words the
conditions under which two different solutions of a triangle may
be expected when two sides and an angle opposite one of them
are given. Illustrate with sketches.
E. Miscellaneous examples on solution by dividing into two
right-angled triangles.
1. There are six parts of a triangle three sides and three
angles. When three of these are known it is generally possible
to calculate the other three and solve the triangle.
52 SOLUTION OP TRIANGLES [CH. IX
The list below gives the possible ways in which three of
these parts may be given.
Copy each item and determine by drawing a sketch (1) whether
the triangle can be solved, and if so (2) the first step in the
solution :
(I) 3 sides (0, 6, c).
(ii) 2 sides and the included angle (a, 5, C).
(iii) 2 sides and an angle not included (a, 6, A),
(iv) 2 angles and the side included (A, B, c).
(v) 2 angles and a side not included (A, B, a),
(vi) 3 angles.
In which cases may two different solutions possibly occur ?
2. Solve the following triangles by drawing a perpendicular:
(1)
A =
54
23',
6 =
19-45, c =
23-51.
(2)
A =
117'
3 21',
B =
35
24',
C =
467.
(3)
A =
41
41',
b =
26-7
,
a =
20-63
(4)
a
4-35,
6 =
7-91
9
c =
6-41.
(5)
B =
63
26',
6 =
114,
a
91-5.
(6)
B =
24
8',
C =
79
53',
a
3-47.
(7)
c =
47
19',
a =
191,
6 =
179.
(8)
a =
13-5,
6 =
14,
c =
16-5.
(9)
c=37 a
23',
o =
91,
a =
124.
(10)
C=
71
35',
A =
52
13',
6 =
7-47.
(11)
a =
147
6 =
92,
c =
69.
(12)
A =
51*
29',
B =
79
47',
a =12-39.
Note. The problems in Chapter XIII may all be solved by the process
above, but the general formulae in Chapters X, XI, XII will be found t*
shorten the work considerably.
CHAPTER X
OBTUSE ANGLES
1. In the figure P revolves round the circle whose centre is
O and whose radius is 10 cm.
AO= 17 cm. PN is drawn per
pendicular to AOQ. / *\P
Calculate the lengths of ON
and AN when the angle POGl
has the values : 36 52', 53 8', A ; 6 FT?**
60, 120, 126 52', 143 8' and \ /
arrange the results in a table.
It will be seen that when *** *""'
POO is obtuse, AN is less than pjg 22.
17 cm.
But in the figure as drawn, AN = 17 + 10 cos POGl If this
is also true when POGl is obtuse, what do you conclude with
regard to the value of the cosine of an obtuse angle ?
2. For which pairs of the angles above is the length of ON
the same? What is the sum of each pair? Write down the
values of cos 120, cos 126 52', cos 143 8' and express in words
the relation between the cosines of supplementary angles.
3. Find the length of PN for each of the angles in question 1.
For which pairs is the length of PN the same? It is customary
to say that PN = 10 sin POQ for all values of the angle POGl acute
or obtuse.
Write down the values of sin 120, sin 126 52', sin 143 8'.
Note that PN is drawn above the line in every case, bat ON is drawn
from O in one direction for an acute angle POQ, bat in the opposite direction
for an obtuse angle POGL
54
OBTUSE ANGLES
[CH. X
4. The figure shows that 1
eosll2'37'=:- T V
Calculate the length of the
perpendicular (Pythag.), and
write down the values of the
other five ratios of this angle,
paying special attention to
sign.
5. From the Tables find
the values of the sine and co-
sine of : 140, 123, 104 13',
166 51', 152 22'. u Fig. 23.
6. Find the tangent of each of these angles. Draw a rough
sketch to determine the sign.
7. Find the cotangent, secant and cosecant of each of the
angles above.
8. Find the value of
sin 64, sin 164, cos 53 30', cos 153 30',
sin 101 25', cos 121 33', sin 111 11', cos 174 19',
tan 41 23', tanl4123', cot 19 21', cot 99 2V,
cosec3141', sec 12 17', sec 131 41', cosec 112 17'.
9. Find two angles less than 180, whose sine is 0-7660.
10. If A is an angle of a triangle, find A if
sin A = 0-9. cos A = - 0-6225, sin A = 0-2860,
cos A = 0-1225, cos A = - 0-1225, sin A = 0-3333,
tan A = 2-6051, tan A = - 2-6051, cot A = -0-2401,
cosec A = 1 -1383, sec A = 3-1085, sec A = - 3-1085.
11. In dealing with triangles any angle between and 180
may occur. What can be asserted with regard to the sine of
any angle of a triangle ? What can be asserted with regard to
the cosine of any angle of a triangle ] These results should be
carefully remembered.
CHAPTER XI
THE SINE FORMULA
1. In the triangle ABC having A = 40, B = 80 and a = 10 cm.
calculate C, 6 and c by drawing a perpendicular, and show that
__ sin B
a ~~ sin A *
n oi. i xi_ i. c sinC ,6 sin B .... . .
2. Show also that - = . and - = . in this particular
a sin A c sin C r
triangle.
3. Draw any acute-angled triangle at random (i.e. without
making the angles and sides any chosen size or length). Measure
the sides and angles and show that = -: = .
sin A sin B sin C
4. Explain clearly in words the facts verified above.
5. Consider whether the rule is true for obtuse-angled
triangles. Verify by drawing such a triangle at random and
carefully measuring its sides and angles. (Remember the fact
with regard to the sines of supplementary angles which was dealt
with in Chapter X.)
6. Making use of the sine rule above, calculate a in the
triangle having A = 62, C = 53 8', c = 8cm., and verify by
accurate drawing.
56 THE SINE FORMULA [OH.
7. Calculate the following and verify by accurate drawing.
Note. It is convenient to write the required part as the numerator of a
fraction.
(1) A = 63, C = 50, c = 6 cm., find a.
(2) B = 47, C m 73, c = 11 cm., find b.
(3) C = 30, A = 81 30', a= 6 cm., find a
(4) A = 44, B = 3r, a=4cm., find 6.
(5) B = 39 24', C = 49 12', 6=5 cm., find c.
sin A
8. If 10 sin A = 3 sin B, what is the value of -; - ?
sin B
i , .. e sin C . T 7 ,, , a c
And rf a = ETA ' show dearly thafc =
9. To prove the Sine Formula for an acute-angled triangle
draw a sketch of any such triangle ABC. Draw ON perpendicular
to AB. In the triangle CAN express CN in terms of b and A ; find
also an expression for CN in terms of a and B in the triangle
BAN. By equating these prove that -. - - - for any acute-
sin A sin o
angled triangle.
Note. An alternative proof will be found in the Appendix.
10. Bemembering that sin (180 - A) = sin A, prove the sine
formula for any obtuse-angled triangle.
11. Use the Sine Rule to find the following :
(1) If A = 36 52', B 30, a = 10 cm., find 6.
(2) If A 58 13', C = 23 35', c = 4 cm., find a.
(3) If B = 68 26', C 63 8', c = 3-5 cm., find b.
Note. Much time may be saved by Baying "a over b equals sin A over
tin B " and at the same time writing a* b x
XI] THE SINE FORMULA 57
(4) If A = 5116 / , C = 73'49', a-12cm,, findo.
(5) If B = 20 11', A = 59 51', b = 17 cm., find a.
(6) If A = 43 32', B = 61 39', c = 10 cm.; find C and a.
(7) If B = 2113', = 51 42', <* = 4cm., find 6.
(8) If A = 44 32', C = 67 21', 6 = 5 cm., find a.
(9) If A = 21 6', 6 = 10, a = 6, find two values for B.
(10) If C = 32 37', c 3-6, 6 = 5, find two values for B.
(11) If B = 37 23', 6 = 2, a = 2 -5, find A. Are there two values t
(12) If B = 37 23', 6 = 3, a = 2'5, find A. Are there two values ?
12. If in Fig. 1, p. 2 the angle A=24, ^BPC = 54' and
AP SB 10 cm., calculate L ABP, BP (by the sine formula) and hence
find BC.
13. In the same tig. if A = 41, L BPC = 61 and AP = 200 feet
find a trigonometrical expression for BC without working with
tables. By substituting the values given in the tables calculate
the length of BC.
14. Use the method indicated above to solve the following.
The angle of elevation of the top of a tower is observed to be 25
and from a point 100 feet nearer it is 43. Find the height of
the tower.
15. From a boat the angle of elevation of the top of a cliff
200 feet high is found to be 29. The boat sails directly towards
the cliff. How many feet must it sail so that the angle of eleva-
tion may be 50* 1
16. From the top of a cliff the angles of depression of two
buoys in a straight line and 124 feet apart are 18 47' and
51 29'. How high is the cliff?
Nott. Questions B 6 and several in B 2 of Chapter IX may be solved by
means of the sine formula.
CHAPTER XII
THE COSINE FORMULA
1. In the triangle ABC if b - 3 in., c - 4 in. and A = 90, show
that a must be 5 in. long.
If A is acute, will a be greater or less than 5 in. 1
If A is obtuse, will a be greater or less than 5 in. ?
To find a relation between the length of a and the size of the
angle A in this triangle, draw an accurate figure with A = 20, 40,
60, 80 in succession, and measure the length of a in each case.
Verify that 6 a + c a a 3 = 2bc cos A in each case. Arrange results
in a table.
2. By drawing a perpendicular CN from C to AB in each
case above and calculating the lengths of AN, CN, NB, a, verify
that 6 2 + c a - a* = 2bc cos A.
3. In Chapter IX, question A 2, it was found that if
6 = 10 cm., c = 9 cm. and A = 60, then a = 9-539 cm.
Verify the formula above by substituting these values.
Note. This formula is very useful and should bo remembered. A formal
proof will be found in the Appendix.
4. Verify the following by means of the Cosine Formula :
(1) If b = 7-5 cm., c = 9cm., A = 53 8', then a = 7-5 cm.
(2) If = 10cm., c= 12 cm., A = 23 35', then a = 4*903 cm.
(3) If b = 25 cm., c = 24cm., A = 41 25', then a = 17-35 cm.
CH. XIl] THE COSINE FORMULA 59
5. If in any one of the figures above the letters denoting
the sides and angles were altered so that A became B, B became
C, a became 6, etc., then what relation between the sides and
cos B would be true ?
Write down the cosine formula in a triangle ABC in three
different forms, thus :
a 2 = 6 2 + c a -..., 6 a = ............ cosB,
6. If the angle A is a right angle, what is the value of cos A,
and what does the formula become 1
Observing that cos B = - in this triangle, show that the formula
is true in each of the three forms given in question 5.
7. Consider whether the same formula is true in an obtuse-
angled triangle.
To do this calculate the length of a in a triangle ABC having
6 = 8 cm. and c = 10cm. when A is 90, 110, 130, 150, 170.
(Draw CN perpendicular to AB produced.) Arrange results in a
table and (remembering the important fact about the cosine of
an obtuse angle) verify that a* = 6 a + c a - 26c cos A.
8. Verify the cosine formula in each of the following :
(1) If 6 = 10 cm., c=7-5cm. and A = 126 52', then a =15-69 cm.
(2) If a = 19cm., c = 45cm. and 8 = 161*47', then b = 63 -33 cm.
9. Find the third side in each of the following triangles :
(1)
(2)
(3)
(4)
a = 5,
6 = 3-5,
6 = 3-5,
c
c
= 4,
= 4,
= 10
= 10
C = 32.
C = 148*.
, A = 71 36'.
, A = 108 24'.
<*)
a = 6-25,
c - 3-2, 1
3 = 21
13'.
(6)
a =17-3,
6
= 23
4, i
C = 34
25'.
(7)
a=17-3,
6
= 23
4, (
= 145
35
/
60 THE COSINE FORMULA [CH. XII
10. By substituting in the cosine formula, find the angles of
the following triangles. Verify by accurate drawing.
(1) o = 4 in., 6 = 5 in., c = 6 in.
(2) o = 5 in., 6 = 6 in,, c = 7 in.
(3) o = 4 in., b = 5 in., c = 3 in.
(4) o=7'5 cm., b = 4*4 cm., = 4-3 cm.
11. If the sides of a triangle are 3, 5 and 6 inches long,
calculate the smallest angle.
12. Find the largest angle in the triangle whose sides are
4, 5, 2 inches long.
13. Repeat question E 1, Chap. IX, showing how each of the
required parts may be found by using either the cosine formula
or the sine formula.
14. Two trains start at the same time from the same station
along straight tracks making an angle of 40. If their average
speeds are 20 and 24 miles an hour, how far apart are they in
half an hour ?
15. What angle does a straight rod 29 feet long subtend at
the eye of an observer who is 18 feet from one end and 25 feet
from the other ?
Note. Questions 4 and several of those in E 2 of Chapter IX may be
solved by means of the cosine formula.
CHAPTER XIII
PROBLEMS
Note. An example of orderly arrangement will be found in the Appendix.
1. From the ends of a breakwater a mile long and running
E. and W. the bearings of a certain buoy are found to be
S. 27 W. and S. 72 E. Find the distance of the buoy from
each end in yards, and also its distance from the nearest point
on the breakwater.
2. From the same breakwater the bearings of another buoy
are S. 63 53' W. and S. 41 19' W. Find its distance from each
end.
3. From the E. end of the same breakwater a point on shore
is found to be 1500 yds. N. 15 28' W. Find the distance of this
point from the other end.
4 If A is 6-4 miles S. 17 35' E. of P, and B is 4-3 miles
S. 31 17' W. of P, what is the distance AB ?
5. A balloon is observed from two points 1250 feet apart on
a level plain at the moment when it passes above the line joining
them. If the angles of elevation are found to be 39 51' and
61 23', find the distance of the balloon from each point and its
height above the plain,
6. A vertical wall runs horizontally across the side of a hill
which slopes at an angle of 10 25' with the horizontal From a
point down the slope 110 feet from the base of the wall the angle
of elevation of the top is found to be 15 57'. Find the height
of the wall.
62 PROBLEMS [CH.
7. The angles of elevation of the top of a certain tree from
two points on opposite sides of it are found to be 52 31' and
63 P 5'. If the points are 125 feet apart, find the height of the tree.
8. In a certain steam engine the crank OP is 18 inches long
and revolves about O. The other end C of the connecting rod
CP (which is 72 inches long) moves backwards and forwards
along a straight line passing through O, Find the angle at C
and the distance CO when the angle COP is
(1) 24, (2) 90, (3) 124.
9. Find the angle at O and the distance CO in the same
engine when the angle at C is (1) 13, (2) 10 23'. (There are
two answers to each.)
10. A tower stands on a slope which is inclined at an angle
of 17 Q 15' with the horizontal. From a point further up the
slope and 370 feet from the base of the tower the angle of
depression of the top of the tower is found to be 9 38'. Find
the height of the tower.
11. From a ship sailing N. 22J W. a lighthouse bears
N. 31 '3 E. After the ship has sailed 12 miles the lighthouse is
found to bear due E. Find its distance from the ship at each
observation.
12. P and Q are points at opposite ends of a wood. From
a point R outside the wood the angle PRO, is observed to be
79 47', and PR = 272 yds. and QR = 349 yds. Find the distance
PCt in yards.
13. In the triangle ABC having B = 49* 52', C = 61 34',
a = 17*4 cm., the bisectors of the angles meet at I. Find the
radius r of the circle which touches AB, BC and CA.
(If ID is perpendicular to BC, it is a radius. Find BD in terms of r,
CD in terms of r.)
11 In the triangle ABC, A = 43, a = 12-5 cm. What angte
does BC subtend at the centre of the circle passing through A, B,
C) Calculate the radius of the circle. If 6 = 97 cm., find the
angle which it subtends at the centre.
XIIl] PROBLEMS 68
15. A tower subtends an angle of 34 at a point P which is
80 yards due East of it. What angle will it subtend at a point
Q which is 60 yards due South of P ?
16. A ladder 20 feet long makes an angle of 30 with the
wall of a house. How much nearer to the wall must the foot of
the ladder be brought in order to reach 18 inches higher than
before, arid what angle will the ladder now make with the wall ?
17. A vessel starts from O, sails 4 miles N. 38 E. to P, and
then 6 miles S. 16 E. to Q. She then returns direct to O. How
far and in what direction has she to go, and how near will she
pass to P?
18. From the top of a wall 5 feet high the angle of elevation
of the top of a flagstaff is observed to bo 63 31', while the angle
of depression of the foot is 5 42'.
Find (a) the distance of the foot of the wall from the foot of
the flagstaff, (6) the height of the flagstaff.
19. A straight road runs N.E. for 300 yards from A to B,
From A a tower bears N. 19 27' E., and from B, N. 70 33' W.
Find the distance of the tower from A.
20. Solve the triangle in which a = 28*3 cm., c = 12*5 cm. and
B = 32,
21. In a triangle B = 40, C = 70, a = 123 yards. Find the
side c and also the area of the triangle,
22. In a triangle a = 5 in , 6 = 7 in., c = 9 in. Find the
greatest angle.
23. The length of a breakwater which lies along a meridian
is 789 yards. From a boat the Southern extremity bore
S. 67 30' W. and its distance was 426 yards.
Find (i) the distance of the boat from the breakwater, (ii) the
bearing and distance of the boat from the Northern end of the
breakwater.
64 PROBLEMS [OH.
24. Two ships leave a port at the same instant. . One steams
& 11 16' E. at 15 knots, and the other S. 22 30' W. at 19 knots.
Find their distance apart at the end of half an hour and the
bearing of one from the other.
25. A parallelogram ABCD has AB=3-64", 80 = 5-82", and
the angle 8 = 67. Calculate the length of the perpendicular
drawn from A upon BC and also the angles which the diagonal
AC makes with the sides of the parallelogram.
26. Two tangents to a circle of radius 5 cm. from an external
point are each 8 cm. long. What length of arc is included
between the tangents?
27. The sides of a parallelogram are 12" and 8", and include
an angle of 37 32'. Find the area of the parallelogram, and the
lengths of its diagonals.
28. A ship S at sea finds the bearings of a lighthouse L and
a buoy B to be N. 32 W. and N. 14 E. respectively. B is found
to be 3*5 miles N. 66 E. from L. Find the distances of the ship
from the lighthouse and the buoy.
29. A ladder reaches a window ledge 26 feet above the
ground on one side of a street, and makes 70 with the ground.
Find the length of the ladder.
On turning the ladder over without moving its foot, it is
found that when it rests against a wall on the opposite side of
the street it makes 20 with the ground. Find the width of the
street.
30. Two lighthouses are known to be 8f miles apart. A
ship observes their bearings to be S. 41* 48' W. and S. 19 T W.
respectively* After making due South for some time she observes
that they are both in a straight line due West. How far has
she sailed 1
31. A, B are two points 1200 feet apart on the straight bank
of a river flowing due East. A point C on the opposite bank
bears N. 53* 19' E. from B and N. 24J W. from A. Find the
perpendicular width of the river at (X
XIII] PBOBLEMS 66
32. In a circle of radius 3 cm. an isosceles triangle ABC is
described with base BC 3*5 cm. long. Solve the triangle ABC.
33. The diagonals of a rhombus are 13 cm. and 9 cm. long.
Calculate the length of the radius of the circle inscribed in the
rhombus.
34. For a quick survey of a bay a ship observed two points
P and Q at sea level.
Bearing of P, S. 73 W.
Q, S. 87 E.
At P angle of elevation of masthead is 3.
^ > >i
Hc3ight of masthead above water-line 120'.
Find (i) distance of P from ship, in yards ;
(ii) length of base line PQ, in yards.
35. (i) Solve a triangle ABC, given AB = 23'6 in., BC = 18*5 in.,
angle A~ 28. Arc there two possible solutions'?
(ii) Solve a triangle ABC, given AB = 62 yards, AC = 83
yards, angle A = 76 28'. Are there two possible solutions?
36. Two rocks A and B lie due East and West of one another
and are seven miles apart. From a ship A bears S. 24 W. and
B bears S. 35 E.
How far is each rock from the ship ?
37. The flagstaff outside a certain coastguard station is
24 feet in height and stands at the summit of a perpendicular
cliff. From a boat the angles of elevation of the top and bottom
of the flagstaff are observed to be 30 63' and 26 44' respectively.
Find the distance of the boat from the foot of the cliff.
38. Two circles of radii 3 feet 6 inches and 2 feet 3 inches
have their centres 4 feet apart. What is the angle between their
common tangents?
p. T.
66 PBOBLBMS [CH.
39. P and Q are two points on a field such that Q is 66 yards
N. 38 15' E. from P. From Q a tree bears N. 74 21' W., and
from P the same tree bears N. 24 8' W.
(i) Find the distance of Q from the line joining P to the
tree,
(ii) Find the distance of the tree from Q.
40. In a triangle ABC, B = 110, C = 20, a = 158 ft.
Find the side 6, and the area of the triangle.
41. O is the centre of a circle, and AB a diameter 10 inches
long. C is a point on the circumference. The straight line AC
is 7*66 inches long. Find (i) the angle AOC, (ii) the length of
the minor arc AC.
42. The position of an inaccessible point C is required.
From A and B, the ends of a base-line 220 yards long, the
following bearings are taken :
From A the bearing of B is N. 70 30' E. and the bearing
of C is N. 30 20' E. From B the bearing of C is N. 59 40' W
Find the distances of C from A and B.
43. From a ship steaming N. 50 E. a lighthouse bears
N. 30* E. Six minutes later the lighthouse bears N. 10 E.
When will it be abeam and how near will the ship approach it ?
Speed of ship 10 knots.
44. In a simple steam engine the connecting rod CP is
15 feet 9 inches long and the crank 3 feet 4 inches. Find the
angle at P when the distance CO is 17 feet.
Fig. 24.
XIIl] PROBLEMS 67
45. A rugby football ground is 76 yards wide, and the goal
posts are 18 ft. 6 in. apart. Find the angle subtended by the
line joining the goal posts at (i) the middle point of the 25-yard
line, (ii) the point where the 25-yard line meets the touch line.
46. Two points A, B are taken on a level plain such that the
distance between them is 1 mile. A point C on the same level is
observed from A and B, and angles CAB and CBA are measured.
If CAB = 40 and CBA 100% calculate the angle ACB and the
lengths of CA and CB in miles.
47. If with the same base-line (AB) as in the last question a
point D is observed and DAB = 80 and DBA = 70, calculate AD
and DB.
48. What is the angle CAD in the figure of questions 46 and
47 ? By means of the triangle CAD, find the length of CD.
49. By means of the triangle CBD, find the length of CD.
(The answer should be the same as that of the last question.)
50. Using the results of the questions 46 to 49 above, show
how CD can be found in the shortest way when AB = 1 mile and
the angles at A and B are the same as before. Arrange the work
very neatly and verify by drawing to scale.
51. A base line AB is 2 miles long ; C and D are two points
on the same level as AB and the angles at A and B are observed.
Find CD if ABC - 76, ABD = 54, BAC=*43, BAD =113*.
Arrange the work as in the last exercise.
52. In the triangle CAD of the last question calculate the
angle ADC.
Now of A is known to be due West of B, find the bearing
of C from D.
The process above is called Triangulation and is used in
making maps of the country.
62
68
PBOBLEMS
[CH.
53. P is 800 yards due North of Q. If X is N. 73 25' E. of
P and N. 39 41' E. of Q; and Y is S. 79 20' E. of P and
N. 77 12' E. of Q, find the distance and bearing of X from Y.
Verify by drawing to scale.
54. The angle of elevation of a tower 100 feet high due
North of an observer was 50. What- will be its elevation when
the observer has walked due Bast 300 feet 1
55. The elevation of a balloon was observed at a certain
station to be 20 and its bearing was N.E. At a second station
4000 yards due South of the former one its bearing was N. 1 1 E,
Find its height.
56. The elevation of the top of a spire at one station A was
23 50' and the horizontal angle at the station between the spire
and another station B was 93 4'. The horizontal angle at B was
54 28' and the distance between the stations 416 feet. What
was the height of the spire ?
57. A ship was 2640 yards due South of* a lighthouse.
After the ship had sailed 800 yards N. 33 W. the angle of
elevation of the top of the lighthouse was 5 25'. Find its height.
58. If the jib of an ordinary crane is 15 feet long, the post
is 10 feet high and the tie is 7 feet long,
what angles do the jib and tie make with
the post ?
59. What must be the lengths of the
tie in the crane above when the jib makes
angles of 20, 40, 55 with the post]
60. What are the lengths of the di- Fig. 25.
agonals of a parallelogram having adjacent '
sides 3 in. and 5 in. long and the included angle 40 1
XIII]
PROBLEMS
69
61. If a parallelogram has two adjacent sides 10 cm. and
12 cm. long and the included angle 73, what is the length of the
diagonal passing through the intersection of the given sides ?
62. If the included angle were 107 (instead of 73), what
would the length of the diagonal be ?
63. From one corner O of a cube distances OA, OB, OC are
measured along the edges so that OA=2 in., OB = 3 in. and
OC = 4 in. Find the angles of the triangle ABC.
64. From one corner A of a regular tetrahedron (see Fig. 9)
distances AX = 5 cm., AY = 4 cm., A2 = 3 cm. are measured along
the edges AB, AC, AD. Find the angles of the triangle XYZ.
65. From a point A due West of a hill TN the angle of
elevation of the top T is found to be 43 26'. From B which is
300 feet W. of A its elevation is 27 39'. Find its height.
Fig. 26.
66. If C is 500 .feet 8.E. of B in question 65, what is the
angle of elevation of T from C ?
70
PROBLEMS
[OH.
67. A path AB straight down the steepest slope of a smooth
hillside makes an angle of 40 with
the horizontal. If another straight
path AC starting from the same point
goes down the slope and makes an
angle of 30 with the first path AB,
what is its angular slope L AGO?
AB
Find the ratios
AD
AC , , AC
AB and hen e AD
6
Fig. 27.
and ZACD.
68. If AB in the last question runs due South, in what
direction (give compass bearing) does AC run?
Note that angles CAB, CDB are not equal. CDB gives the bearing.
69. A hillside slopes towards the South at a gradient of 1 in
2. In what direction (compass bearing) must a path be made up
the hillside so that its gradient may be 1 in 5 1
70. If a hillside slopes at an angle of 42 towards the North,
what is the slope of a road running up it in a direction S. 73 E. 1
71. A weight R is suspended by three strings OR, OP, OQ
knotted together at O. If a parallelogram OBCA be drawn (as
in the figure) with CB, CA parallel to PO, QO, and the diagonal
OC in the same straight line as OR, then the tensions on the
strings OP, OQ, OR are proportional to the lengths of OA, OB, OC.
If the weight R is 10 Ibs. and the strings OP, OGl make angles
of 30 and 70 with OC, show that the tensions on these strings
are approximately 6 '08 and 9 -54 Ibs. weight
0///^////////^
Fig. 28.
XIllJ PROBLEMS 71
72. If R = 10 Ibs. and each of the strings OP, OQ makes an
angle of 35 with OC, what must the tensions on OP and OQ be?
73. If R=12 IbS. and tho tension on OP is 8 Ibs. weight
when L POQ = 120, what must be the tension on OQ, and what
is tho angle QOR?
74. What must the weight R be if the tension on OQ is 3 Ibs.
weight, the tension on OP is 5 Ibs. weight, and the angle POQ is
130 1
75. A station A is 1000 yards due West of another station
B and on the same level. A point C is N. 20 E. of B and
N. 50 B. of A and its elevation from B is 30. Find the height
of C above the level of AB and the angle subtended by AB at C.
APPENDIX A
DEFINITIONS AND RELATIONS BETWEEN RATIOS
I. If from any point P in one arm of an angle A a straight
line PN be drawn perpendicular to the other arm, then
PN
the ratio is called the sine of the angle A,
AN
~ ,, cosine ,,
PN
tangent
AP
^j secant
AP
cosecant
AN
cotangent
A N
II. From the definitions above:
(1) cosec A = . secA = , cot A =7 .
^ f sm A cos A tan A
PN
/rt . sin A AP PN . . cos A
(2) = . = -~ = tan A. Also-: - = cotA.
v ' cos A AN AN sin A
AP
. 9 /PN\ 3 /AN\ a PN 2 4-AN 2 ,AP 2
/O\ cin" A J. rrk A i i 4- I 1 -
\ / olll r\ T OLIft r\ I I ~ I I ~ "" -". o ~" "I _Q
by Pythagoras' Tlieorem. c
/. sin a A + cos 9 A SB 1.
APPENDIX A
73
TIT. If ABC is a triangle with a right angle at C, the angles
at A and B are complementary and B = 90 - A.
sin B = = cos A,
tan B = -= cot A,
BC
sec B = = cosec A.
Each ratio of an angle = the co-ratio of its complement.
(90
IV. If an angle POX be continually increased by keeping
OX in a fixed position and revolving OP, PN increases and
ON decreases until POX becomes 90, and then PN = OP and
ON vanishes.
M O N
If OP continue to revolve so that POX becomes an obtuse
angle, PN will begin to decrease and it is measured upwards from
OX as before, but ON will begin to increase and is now measured
in the opposite direction away from the angle.
Thus the sine of an obtuse angle (QOX) is the same as the
sine of its supplement (POX), while the cosine of an obtuse angle
is always negative but its measure is the same as the cosine of its
supplement.
APPENDIX B
SINE FORMULA
Let ABC be any triangle and O the centre of the circum-
scribing circle.
Draw a diameter BOAj and join AjC.
The angles at A and Aj are equal (in the same segment).
If R is the radius of the circle, then BA X = 2R.
Also BCAj = 90 (in a semicircle).
BC a
.*. = sin BAjC or ^-~ = sm A,
BA &r\
that is 2R = .
sin A
By joining AAi it may be shown in the same manner that
2R=-4-.
smC
By drawing a diameter through A (or C) it may also be shown
that
8inB
a
* i
in any triangle ~. - =
sin A sin B
c
sin C
APPENDIX C
COSINE FORMULA
Let ABC be any triangle with an acute angle A.
Draw BN perpendicular to AC. Let its length be p units.
Suppose AN is x units long. Then AC is b x units.
Then a 2 = p* + (b - a:) 2 (by Pythag.)
= 6 2 + c a - 2bx (for c* = j
= 6 2 -f c 2 26c cos A (for 05 = c cos A).
If the angle A he obtuse, with the same construction as before
ON will be 6 + x units long.
Then a* = p* + (b + xf (by Pythag. )
a; (for e = p 1 + a?)
26CCOSBAN
. ^ + c a _ 25 C cos A (for cos BAN = - cos A).
APPENDIX D
EXAMPLES OF ORDERLY ARRANGEMENT OF WORK
I. Solve the triangle ABC in which
C = 90
C = 90 }
A = 37 16' V.
a= 2-1 cm. J
B = 90 -37 16' ~ 62* 44'.
6 = a tan B = 2-1 x 1-3143
= 2 -7 60 cm.
acosec A = 2-1 x 1-6515
= 3-468 era.
2'lcm,
13143
21.
26286
13143
276003
16515
21
33030
16515
346815
APPENDIX D
77
II. Solution by dividing into right-angled triangles.
Giveu A = 44 23' j
6 = 7-6 cm. L
c = 5 cm. )
Draw BN perpendicular to AC.
AN = 5 cos 44 23'
= 5 x 0-7 147 =3-5735.
/. CN = 7-6-3-5735 = 4-0265 cm.
Also BN = 5 sin 44 23'
= 5 x 0-6994 = 3-497.
n .
But
But
BN
= -j
= 0-8688.
= 40 59'.
A = 44 23'.
3-497
3-497
4-026
0-5437
0-6048
T-9389
(Check A+B-f = 180 0'.)
CB =
12-23 + 16-21 = V2*R4
: 5-33 cm.
!a- 5-33 cm.
B - 94 38 .
0=40 59'.
78
APPENDIX D
III. Solution by formulae.
Given A = 64 23'
6 = 6-31 cm.
= 5*27 cm.
= 6-3P + 5-27 3 -2 x 6-31 x 5-27cos6423'
= 39-82 + 27-77 -28-75
-67-59-28-75
= 38-84.
/. a = 6-232 cm.
sin A 5-27 x sin 64 23'
2
6-31
5-27
cos 64*23'
Also
sin C = c -
6-232
.*. C = 49 42' or its supplement.
But C must be acute, for c is not
the longest side.
5-27
sin 64 23'
6-232
But
(Check A
0-49 42'.
A = 64 23',
B = 65 55'.
= 180 0'.)
0-3010
0-8000
0-7218
1-6358
1-4586
0-7218
1-9551
0-6769
0-7946
1-8823
Ja = 6-23cm.
<B = 65 55'.
*C=4942'.
ANSWERS
The lengths have in general been calculated to four significant
figures and the angles to the nearest minute. Slight discrepancies
will often appear if the methods of calculation differ from one
another.
CHAP. I
1. 180, 45, 30, 6, 18. 2. 22 30', 7 30', 33 45', 64 17'.
3. 53 25', 77 0', 58 34', 139 46'.
4. 67 50', 58 45', 60 35', 42 25', 51 C 9', 70 17', 10 3', 30 Q 11'.
5. 126 35', 103 0', 121 26', 40 14'. 6. 60, 23% 46 39'.
7. 36, 72, 72; 20, 80, 80; 45, 45, 90.
8. 72, 108, 72.
9. (1) 45, 135, 45. (2) 40, 140, 140. (3) 51f .
10. S. 31 W. 11. 90. 12. N. 13* EL
13. S. 45 B. 14. 24. 15. 37.
16. 5. 17. 60, 30. 18. |, *, .
19. ABP = 48. 20. x-y\
21. 60, 60, 90, 30. CK = 5cm. AK = 8*66 cm AKN is
greater than ACN. AKN decreases.
22. It varies, and is greatest when equidistant from the two
points.
23. The circumference of the circle passing through the ship and
the two points.
OHAP. II
13. 1-4826, 1-5051, 1-5108, 1-5080;
0-5317, 0-5362, 0-5384, 0-5373;
0-2946, 0-3476, 0-0673.
15. 25% 72, 17 30', 55" 12', 21 55', 74 50', 11 10', 66 39',
50" 9', 33 33'.
10. tan 35 = 0-7002. 17. 38" 40', 63 26', 74 3'.
80
ANSWERS
18.
20.
22.
25.
28.
31.
33.
34.
35.
36.
37.
38.
39.
40.
43.
46.
48.
50.
55-43 feet.
67 23', 22 37', 13 feet.
134-9 feet.
5-139.
2-690.
A = 38 40',
A = 34
19. 47-67 feet.
21. S. 35 32' B., 8-602 miles,
30 58'. 24. 5-095.
5-154. 27. 1-288.
127-5. 30. 38 40'.
32. A = 58, B = 32.
21',
B = 47 43',
A = 00 49',
B = 767', 6-42-08, c=43-34.
A =17 19', a = 1-269, c = 4-262.
= 36 52', c = 15-5.
B = 5018', c= 194-9.
41. 248-6 feet.
44. 167-3 feet.
47. 1-82 cm.
23.
26.
29.
B = 5120'.
B = 55 39'.
a =23-10, c
6=0-9048,
= 34-34.
o = l-856.
A = 53 8', B
A = 39 42',
54 feet.
69-88 yards.
11 2 37'.
42.
45.
1003 feet.
28-8 in.
88-18, 59-62 feet.
49. 451-3 yards, 12 48'.
3341'
51. 42-88 sees. 52. 193-0 feet.
54. 93-26 feet. 55. 57-82 feet.
57. S-165 cm., 54 44', 70 32'.
53. 396-2 feet,
56. 45, 54 44'.
ANSWERS
81
11.
13.
14.
16.
CHAP. Ill
0-5764, 0-5779, 0-5771, 0-5766; 0-9559, 0-6839, 0-1888,
0-9898.
25, 71, 54 30', 16 48', 16 51', 78 23', 75 57', 46* 4'.
30. 15. 23 35', 48 36', 25 28'.
4-54, 4-571, 4-581, M45, 8-416, 2-210.
17.
6=67-10, a = 33
51, A =26 32'.
18.
(1) B
= 17 46',
a = 20, 6 = 6-407.
(2) A:
= 64 37',
a =9-395, 6 = 4-457.
(3) A
= 47 47',
a = 77-91, 6 = 70-68.
(4) B
= 78 41',
a = 3-791, 6 = 18-95.
19.
2 18'.
20. 34'.
21. 46', 1 26'.
22.
23 ft. 11 in.
23. 0-755 in.
24. 4-23 mi.
25.
6-47 in
26. 32 14'.
27. 3 ft 1 in.
28.
(1) 0-771, 1-845.
(2) 1-909, 0-595.
CHAP. IV
2.
0-515, 0-777.
3. 60% 41
4", 68-9*.
6.
0-5592,
0-5534, 0*
5519; 0'552G(7).
7.
0-8809,
0-9592, 0-2866, 0-1417, 0-5999,
0-9764.
8.
(1) 8-91. (2)
7-27. (3) 8-30.
(4) 1-003.
9.
44-65 feet,
10. 14-85 mi.
11. 1736yarda
12.
12-6 feet
13. 4-77 in.
14. 69-09 mi.
15.
1-054 feet.
16. 7-5 cm.
17. 9-34 feet
18.
5 ft. 6-1
. in.
19. 3064 mi., 19260
mi 20. 10-86.
21.
Half the length of the Equator. Half.
CHAP. V
5.
cosec A
= , secA
o . A 6
as T , COt A SB - ,
6 a
6.
Angle
56* 27'
28 14' 16* 25'
73' 2V
cosec
M999
2-1140 8-5383
1-0437
sec
1-8094'
M351 1-0425
3-4903
cot
0-6631
1-8624 3-3941
0-2991
P. T.
6
82 ANSWERS
7.
As angle increases, sec increases, cosec and cot
decrease.
8.
22-65, 33-48,
42-55 ft. 9. 32-86 ft.
10.
1191 ft.
11.
10 21 cm.
12. 11 -33 ft.
13.
47-91 in.
14.
156-7 ft.
15. 6-22 mi.
16.
49-05 in.
17.
18-70 ft.
18. 11-63 in.
19.
1 in 14.
20.
20-27 ft.
21. A = 36 52', B =
53 8'.
22.
A = 77, B = l
3,
23.
sin A = cos B,
cot A = tan B, sec A = cosec
B.
24.
22 37'.
25. 16 16'.
Oft
O*7 4- 01-1 A 2ir>5 A i nricfl A 1
zo,
&/. tan A, sin A 4- cos A A.
cos A
a
00
sin A c a
aO 9
cos A b b
c
a a + c ,
and sin* A -f cos* A = -r -f -r = -=- = = 1.
c 2 c 9 c 9 c 2
30. sin 90 = 1, cos 90 = 0, tan 90 =00 (the nearer 90, the
larger), sin 0* = 0, cos = 1, tan = 0.
32. sine and cosine must be less than 1. Tangent may have
any value.
335. sec 2 A -tan 2 A 1. 36. cosec 2 A - cot 3 A = 1.
CHAP. VI
etc.
a o o
n . x p a .
3. sin A = - = 7 = - etc -
a b c
4. B = 3652', A = 53 8', 6 = 15, aj=16, #-9, c = 25cm.
5. A = 6723', B = 23 37', a = 15 6, #=14-4, y = 2-5, c = 16-9cm.
6. ^>a=ajsina, y = rctana, AC = a3seca; CN = jt? tan a = a; sin a tan a.
7. a cos B = x, b sin A = />, c cos B = a, a cos A = j, p tan B = y,
p cot B = .r, p sec A = a, p cosec A = fc, b sin B = y.
8. x sin ap 9 x tan a = t/, p cot a = AN, y cos a = p, p tan a = CN.
ANSWERS 83
9. A = 3541', 6 = 4-700, c = 1141.
10. B = 60 7', 6 - 34-98, c = 40-34.
11. B - 45 29', 5 = 1-711, a = 1 -683.
12. A = 59 25', B = 3035', c = 28-7.
13. A = 33 22', B - 50 38', c - 1 16-9.
14. B = 1747', a = 48-33, c = 50-75.
15. A = 55 23', a = 987 -5, 6 = 681-6.
16. B = 30, A =60, a = 9-873.
17. A = 11 32', B = 7828', 6=16-42.
18. A =19 28', B = 7032', 6=ax2-828.
19. 821-5 ft. 20. S. 55 47' W. 21. 67-02 ft.
22. 177-5 ft. 23. 057'. 24. 8-274 cm.
25. 2-157 cm. 26. 30 14'. 27. 7-779 cm.
28. 9-398 cm. 29. 8-402 cm., 42 48'. 30. 181-4 yds.
31. 6-824 in. 32. 53 8'. 33. 83-25 in.
34. 3-441 cm. 35. 4-253 cm. 36. 4-330, 5cm.
37. 7-694, 8-090 cm. 38. 8'83 mi. per hr.
39. 54 ft. in., 60 ft. 4-5 in. 40. 69 41'.
41. 36 52', 66 25'. 42. 50 12'. 43. 27 54'.
44. 34 58'. 45. 8-201 mi. S. 52 26' E.
46. 52 0', 68 40', 75 24'. 47. 15-64 cm.
48. 16 42', 17 28'. 49. 20 ft. 1 in. 50. 747 fb.
51. 4650 fath. 52. 90, 66 25', 9-165 in.
53. AO = 11-92, BO = 8-39, AC = 15-56, BC = 13-05 inches.
54. 1-595. 55. 18-03 mi. S. 71 43' W.
56. 5-362, 7-332 mi. 57. 83-39 ft. 58. 1132 ft.
59. 027'. 60. 4-189 mi. 61. 51-5 ft., 354-6 ft.
62. 190ft. 63. 76-9 yds. 64. S. 70 E.
65. 140, 141 ft. 66. 11-18 mi. S. 23 37' W.
67. 5-342 in., 0-395 in.
68. 4-35 mi. N. 38 2# E. or S. 38 28' E. 69. 30 2V.
70. 297/9, 268-0, 29-2 ft. 71. 5 mi. N. 53 8' W.
72. 9-063, 4-226 mi. 73. 2-588 mi.
74. 16-73 knots, 18*99 mi.
75. 2-908 in., C = 68 12', A = 74 56'. 76. 65-8 ft.
84 ANSWEBS
77. 63-74 a 78. 0-515, 0-4415 mi. 79. 111*53'.
80. 30 58'. 81. 3-215 mi. 82. 54-63, 64-97 ft.
83. 1125 ft. 84. 1-884, 5-73, 6-85 mi.
85. 65 23', 53 54', 62 43'. 86. 3-535 in., 45, 54 44'.
87. 59 ; , 25 22', 16 36'. 88. 686-9, 1069 ft., 13 10'.
89. 11 48'. 90. 382yds.
CHAP. VII
1. 50 sq. cm., 13297-68 sq. yds.
2. 25 sq. cm., 6648*84 sq. yds. 3. 7-5, 6, 13*5 sq. cm.
4. Half the product of the number of cm. in BC and AN gives
number of sq. cm. in area.
5. 3-214, 4-50 cm., 11-25 sq. cm.
6. Product of number of cm. in base and number of cm. in
perpendicular height gives number of sq. cm. in area.
7. Area = xy sin A. 8. \xy sin A.
9. 5 sq. cm., 9 sq. cm., 16-9 sq. cm., 118 sq. cm.
10. 28-7 sq. cm. 11. 259-8 sq. cm.
12. 237-8, 273-7, 282-8, 289-3 sq. cm
13. 363-2, 346-4, 327-4 sq. cm, 14. 36 yds.
15. 53 8', 126 52', 4-268 cm. 16. 78 42', 22 36'.
17. Sq. pyramid, 273'2 sq. cm. Tetrahedron, 173*2 sq. oin.
18. Segment, 25-67 - 21-22 = 4-45 sq. cm. 19. 15-3 sq. cm.
20. 89-04 sq. cm.
OHAP. VIII
2. 10* = N/IO = 3-162. 3. 10* = -yiO=N/3 7 162 = 1-779.
4. 10* = 5-623.
7. 3-7 = 10 0668J , 3-75 = 10, 3-758 = 10 OW48 , 7-64 = 10 08881 ,
7-649 = lO " 8838 , 9-8 = 10 0>9912 , 9-805 = lO 4 .
8. 2, 7-6, 2-75, 2-76, 2-757, 6-16, 6-17, 6-167, 4-486, 7-0.
9. 4-318. 11. 8-572, 8-179, 2-999, 6-451, 8-164, 9-936.
12. 2-940. 13. 2-397, 1-263, 3-855 4 , 1-408, 2 752, 3-766.
16. Logs are 2-6660, 1-8986, 1-2119, 2-8581, 1-8581, 3-4934,
4-8825, 5-3109, 1-8664, 2-3294.
ANSWERS
85
17. 710-1, 33, 3050, 888, 278000, 7115, 337-2, 26-92.
18.
(1) 37-62.
(4) 149000.
(7) 804-2.
(10) 317800000.
(13) 463-1.
(16) 155-2.
(19) 1-006.
(3) 1143.
(6) 87000.
(9) 2637000.
(12) 1-375.
(15) 1-395.
(18) 2-867.
(2) 1390.
(5) 9243.
(8) 1023.
(11) 5-690.
(14) 266-1.
(17) 4083.
(20) 9-779.
20. Logs are 2-0835, T-0835, 1-5518, 4-5518.
22. Logs are 2-5378, T-8860, 3-6656, 4-0433, T-9987, 1-9573,
2-6385, 1-7350.
24. (1) 0-003762.
(4) 0-001390.
25. (1) 0-005690. *
(4) 137-5.
26. 3-684.
28. \/0-0()2 = (10 !
29. (1) 0-2.
(4) 0-2.
(7) 0-8747.
(10) 0-1968.
(2) 0-01490. (3) 0-01143.
(5) 92-43. (6) 1-023.
(2) 46-31. (3) 0-003758.
(5) 0-1031. (6) 10-28.
27. 17-88, 4-236, 10-45, 49-91, 6-223.
= 1 Q2-6505 =: Q-04472.
(2) 0-06325. (3) 0-02.
(5) 0-09283. (6) 0-4309.
(8) 0-3849. (9) 0-7830.
MISCELLANEOUS EXERCISES IN LOGARITHMS
1.
1888.
2.
35-62.
3.
6-811.
4.
0-7492.
5.
305-8.
6.
31620.
7.
5-473.
8.
2-290.
9.
0-3098.
10.
0-0003321.
11.
0-08680.
12,
0-3607.
13.
0-0412.
14.
0-03728.
15.
6-725.
16.
6092.
17.
0-1460.
18.
0-6015.
19.
1531.
20.
64-92.
21.
9-177.
22.
165-8.
23.
13-36.
24.
18-84.
25.
40-15.
26.
1-501.
27.
98-88.
28.
0-5399.
29.
96-22.
30.
0-4846.
31.
0-461.
32.
16-13.
33.
10-93.
34.
110-4
35.
0-4066.
36.
1-749.
63
86 ANSWERS
CHAP. IX
A. L 9-539 cm., 65 C 12', 54 48'.
4. 7-5 cm., 53 8', 73 44'.
5. 4-903 cm., B = 54 40', C = 101 45'.
6. Drop perpendicular on to either of the given sidea.
7. (1) c = 17-35 cm., A = 72 24', B = 66H'.
(2) 6=2-51, A = 65 11', = 53 25'.
(3) a = 19-39, B = 7130', C = 8413'.
8. a = 8-888, B = 103, C=17.
10. a = 15-69, B = 3039', C = 2229'.
12. (1) a = 15-92, B- 115 39', C = 26 33'.
(2) c= 129-7, A -102 4', B *= 19 33'.
(3) c = 53-78, A = 20 13', B - 24 31'.
(4) a = 5-977, C = 46 20', B = 3517'.
(5) 6 = -6333, C = 1250', A = 5 23'.
B. 1. C = 67 23', a = 8-666, b = 9-333.
2. AN = 8, NB = 6, NO = 3-333.
3. If AL be the perp., find angle C, AL, BL, CL, CA, CB.
4. Draw perp. from either end of the given side.
5. From B or C. If BN perp., find C, ON, NB, NA, AB, AC,
6. (1) C = 7524', c = 4-925, 6 = 4-458.
(2) C = 4259', a = 23-54, 6 = 22-52.
(3) C = 4150', a=1106, c = 150-9.
(4) B = 12510', 6 = 50-69, c = 40-83.
(5) B = 1954', a = 63-71, c = 79-86.
C. 1. a; = 0-5 cm., A = 55 47', B = 82 49', C = 4124'.
4. (1) A = 44 3', B = 5237', C = 83'20'.
(2) A = 104 29', B = 4634', C = 2857'.
(3) A = 51 19', B = 11029', = 18 12'.
(4) A = 38 13', B = 60, = 81 47'.
(5) Perp. from B, AN = 1 -827. A r = 58 32', B = 93 23',
= 28 8'.
ANSWERS 87
D. 1. B = 48 36' or 131 24', C = 94' 32' or 11 44', o = 13-29
or 2-708.
3. Only from B. 4. B = 30, C = 113 8', c = 18-39.
5. When the side opposite the given angle is less than the
other given side.
K 1. B = 5222', = 73 15', a =19-96.
2. C = 27 15', a = 9-058, b = 5-907.
3. B = 59 24', 120 36', C = 78 55', 17 45', c = 30-44, 9-439.
4. A = 33 19', B - 92 38', C = 54 3'.
5. A - 45 52', C = 70 42', c = 120-3.
6. A =75 59', 5 = 1-462, c= 3-521.
7. A = 70 36', B = 62 5', c = 148-8.
8. A = 51 45', B = 5432', C = 7343'.
9. A = 55 50', 124 10 ; , B = 86 47', 18 27', b = 149-6, 47-72.
10. B = 5612 / , a = 7-104, c = 8-529.
11. A =131 21', B-280', = 20 39'.
12. = 48 44', 6 = 15-59, c = ll-91.
1. Angle
CHAP. X
36 52' 53 8' 60 120' 126 52' 143 8'
8 (5-9999 say) 6 55 6 8cm.
25 23 22 12 11 9cm.
ON
AN
The cosine of each of these obtuse angles is negative.
2. Supplementary angles. O f 5, 0-6, 0*8. Cosine of an
obtuse angle = (cosine of its supplement).
3. PN = 6, 8, 8-66, 8-66, 8, 6 cm. Supplementary angles.
0-8660, 0-8000, 0-5999.
4. Sine = y|, tangent = \ 2 -, secant = T 6 7 , cosecant = |-|,
cotangent = y\.
5. sin. 0-6428, 0-8387, 0-9694, 0-2275, 0-4638.
cos -0-7660, -0-5446, -0-2456, -0-9738, -0-8859.
6. tan -0-8391, -1-5399, -3-9474, -0-2336, -0-5235.
88 ANSWERS
7. cot -1-1918, -06494, -0-2533, -4-2812, -1-9101
sec -1-3054, -1-8361, -4-0720, -1-0269, -1-1287.
cosec 1-5557, 1-1924, 1-0316, 4-3965, 2-1560.
8. 0-8988, 0-2756, 0-5948, -0-8949,
0-9802, -0-5233, 0-9324, -0-9951,
0-8811, -0-7988, 2-8478, -0-1646,
1-9040, 1-0234, -1-5037, 1-0807.
9. 50 and 130.
10. 64 10' or 115 50', 128 30', 16 37' or 163 23',
82 58', 97 2', 19 28' or 160 32',
69, 111, 103 30',
61 28' or 118* 32', 71 14', 108 46'.
11. The sine is positive. The cosine is positive for acute,
negative for obtuse angles.
CHAP. XI
1. C = 60, b = 15-32 cm., c = 13-47 cm.
4. The ratio of two sides = the ratio of sines of opposite angles.
6. 8-830. 7. 6-979, 8-412, 3-034, 2-966, 5-964.
sin A 3
8. _ = _. 9. CN=68mA = asinB.
sin B 10
11. (1) 8-333. (2) 8-50. (3) 4-069.
(4) 14-77. (5) 42-61. (6) 74 49', 7-139.
(7) 1-515. (8) 3-780.
(9) 36 52' or 143 8'. (10) 48 28' or 131 32'.
(11) 49 23' or 130 37'. (12) 30 24'.
12. 30, 8-134 cm., 6-580 cm. 13. 200 . ~ -^~- = 335-5.
' ' sin 20
14. 93-26 feet. 15. 193 feet. 16. 57-82 feet.
CHAP. XII
1. 1-565, 2-572, 3-606, 4-564.
5. a 2 = 6 a + c a - 26c cos A, 6 2 = c 2 + a? - 2ca cos B,
7. 12-81, 14-78, 16-33, 17 40, 17-94.
ANSWERS 89
9. (1) 2-661. (2) 8-655. (3) 9-495. (4) 11-59.
(5) 3-465. (6) 13-39. (7) 38-92.
10. (1) A = 41 25', B = 55 46', = 82 49'.
(2) 44 25', 57 7', 78 28'.
(3) 53 8', 90, 36 52'. (4) 119 5', 30 51', 30 4',
11. 29 55'. 12. 108 13'. 14. 7'76 miles. 15. 83 6'.
CHAP. XIII
1. 1588, 551, 491 yds. 2. 2019, 3445 yds.
3. 1985 yda 4. 4-82 mi. 5. 1119, 817, 717 ft.
6. 11 -Oft. 7. 98-1 ft.
8. (1) 5 51', 88-0 in. (2) 14 29', 697 in.
(3) 11 58', 60-4 in.
9. (1) 64 10', 78-0 in. ; 115 50', 62-3 in.
(2) 46 7', 83-3 in. ; 133 53', 58-3 in.
10. 49-75 ft. 11. 12-97, 11-33 mi. 12. 403yds.
13. 4-54 cm. 14. 9-164, 63 55'. 15. 28 21'.
16. 3-23 ft., 19 48'. 17. 4-877 mi., N. 57 35' W., 3-982 mi.
18. 50-10, 100-6 ft. 19. 270-9 yds.
20. A =127 29', C = 2031', b = 18-9 cm.
21. c = 1 23 yds. Area - 4863 sq. yds. 22. 95 44'.
23. 393-6 yds., S. 32 10' E., 739-4 yds.
24. 5-29 mi., N. 74 26' E. or S. 74 26' W.
25. 3-351 in., 37 18', 75 42'. 26. 10-13 cm.
27. 58-48 sq. in., 7-463, 18*98 in. 28. 3-834, 4-819 mi.
29. 27-67, 35-47 ft. 30. 15-98 mi. 31. 667ft.
32. A = 35 41', B = = 72 9', AB = AC = 5-7 11 cm.
33. 3-7 cm. 34. 763-2, 1203 yds.
35. (1) = 36 47' or 143 13', B - 115 13' or 8 47', 6 = 35-66
or 6-019 in.
(2) a- 91-24 yds., B = 62 11', C = 41 21'.
36. 6-Q89, 7-459 mi. 37. 254-5 ft. 38. 36 25'.
39. 58-48, 76-08 ft. 40. 194 ft., 5240 sq. ft.
41. lOO'^in. 42. 168,142yds.
90 ANSWERS
43. 4 min. 36 sees., 0-642 miles. 44. 106 10'.
45. 14 4', 4 17'. 46. 1*532 mi., 1 mi.
47. 1-879, 1-970 mi. 48. l-210mi. 49. 1-212 mi.
51. 6-76 mi. 52. S. 41 E. to within 5'.
53. PX = 919-8, PY = 1959, QX = 1380, QY = 1973. XY from
triangle PXY= 1217, and from triangle QXY = 1214 yds.
54. 19 9'. 55. 511-2 yds. 56. 278-6 ft.
57. 192yds. 58. 23 4', 57 7'.
59. 6-564, 9-757, 12-37 ft. 60. 3-32, 7-55 in.
61. 17-73 in. 62. 13-18 in.
63. 75 38', 60 3', 44 19'. 64. 69 32', 63, 47 28'.
65. 351-9 ft. 66. 36 28'. 67. 33 50'. 68. S.37W,
69. N. 69 18' E. or W. 70. 14 45' or about 1 in 4.
72. 6-104 Ibs. wt. 73 13-8 Ibs. wt., 144 44'.
74. 3-84 Ibs. wt. 75. 742-3 yds., 28 16'.
CAMBRIDGE: PRINTED B* w. LEWIS, M.A., AT THE UNIVERSITY PRESS