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TRIGONOMETRY REFRESHER
A. Albert Klaf
DOVER PUBLICATIONS, INC.
Mineola, New York
Copyright
Copyright © 1946 by A. Albert Klaf
All rights reserved.
Bibliographical Note
This Dover edition, first published in 2005, is a reprint of the 1956 Dover unabridged republication of Trigonometry Refresher for
Technical Men, originally published by the McGraw-Hill Book Company, Inc., New York, in 1946.
Library of Congress Cataloging-in-Publication Data
Klaf, A. Albert.
[Trigonometry refresher for technical men]
Trigonometry refresher / A. Albert Klaf.
p. cm.
Originally published: Trigonometry refresher for technical men. 1st ed. New York: McGraw-Hill, 1946.
Includes index.
ISBN-13: 978-048644227-3 (pbk.)
ISBN-10: 048644227-6 (pbk.)
1. Trigonometry. I. Title.
QA531 .K53 2005
516.24—dc22
2004059345
Manufactured in the United States by Courier Corporation
44227603
www.doverpublications.com
PREFACE
We stand at the threshold of a greater technological era. It is safe to say that a knowledge of
trigonometry is and will be more useful to the technical man than any other branch of mathematics.
Mechanical, electrical, and terrestrial forces act in various related positions. Airplanes, ships,
projectiles, and planets move in directions that form certain angles with other objects or planes of
reference. Practically every technical field abounds in problems where a knowledge of trigonometry is
essential.
The aim of this book is to present the subject of plane and spherical trigonometry in such a manner
that all who desire rapid comprehension in this useful field will find their work materially facilitated.
This text follows the method effectively employed in the author’s “ Calculus Refresher for
Technical Men.” The modem question-and-answer form used in both books has been found to give at
once a precise and concise way of presenting the subject. It offers the twofold advantage of easy
understanding and ready application, essentials for progress in any field.
In this book the term refresher is used in the title only because of the manner of treatment and not
because of the omission of proofs or other pertinent material.
Section I treats of angles, their measurement, and their relationships. The fundamental
trigonometric functions of an angle are here presented and defined in a new easily remembered way,
based on a “ shadow-and-perpendicular ” concept (the customary definitions are included in the
Appendix). The methods of periodic functions, identities, inverse functions, and trigonometric
equations are explained in minute detail and illustrated by numerous examples solved in the text. The
important subject of logarithms is developed from the beginning in progressive steps. The student
need not be puzzled when asked to find the value of –18 or to solve (0.004739)0.016. The chapter
entitled “Significant Figures—Accuracy in Computations” should contribute much toward the saving
of time in numerical calculations. The theory, construction, and use of the straight slide rule receive
more prominence here than in probably any other book of this type. The application of the slide rule in
the solving and checking of triangles is duly emphasized. The solution of right and oblique triangles
by the use of logarithms completes this section.
Section II consists exclusively of applications of plane trigonometry. Among the important
problems here presented are those dealing with small angles, periodic functions, vectors and vector
quantities, surveying, aerial and sea navigation, belts, helixes, light, and hydraulics. A complete
chapter is devoted to polar coordinates, complex numbers, De Moivre’s theorem, and trigonometric
series.
Section III consists of a comparatively extensive introductory chapter that is in the nature of a
review of spherical geometry. This is followed by a chapter on spherical triangles. The final chapter
stresses the application of spherical trigonometry in the solution of the terrestrial and the astronomical
triangles.
Throughout the book many illustrations serve to elucidate the text. Numerous problems to test the
student’s progress follow each chapter. Answers to odd-numbered problems are given beginning on
page 571.
The subject of trigonometry has been comprehensively covered in this book, which should prove
useful to those who wish to study the subject for the first time with rapidity, to those who wish to
apply trigonometry to various technological fields, and to those who desire to refresh their previous
acquaintance with its intricacies.
The author takes this opportunity to express his gratitude to William I. Powell, mechanical
engineer, Board of Water Supply, City of New York, for having checked the entire manuscript and for
several helpful suggestions. He is particularly grateful to his wife, Mollie G. Klaf, and his son
Franklin, for their appreciable aid in the mechanics of the manuscript and in the checking of proofs.
A. ALBERT KLAF.
NEW YORK, N. Y.,
March, 1946.
CONTENTS
PREFACE
SECTION I PLANE TRIGONOMETRY
I ANGLES AND THEIR MEASUREMENT
II RELATIONSHIPS OF ANGULAR MEASUREMENTS
III CARTESIAN COORDINATES—QUADRANTS
IV FUNDAMENTAL TRIGONOMETRIC FUNCTIONS
V RELATED ANGLES
VI FUNCTIONS OF CERTAIN ANGLES
VII GRAPHICAL REPRESENTATION OF FUNCTIONS—PERIODICITY
VIII FUNDAMENTAL IDENTITIES—IMPORTANT RELATIONS OF ANALYTICAL TRIGONOMETRY OF
FREQUENT OCCURRENCE
IX INTERPOLATION—BASIC, PRINCIPAL, AND GENERAL ANGLES—INVERSE-FUNCTION NOTATION
X TRIGONOMETRIC EQUATIONS
XI LOGARITHMS
XII SIGNIFICANT FIGURES—ACCURACY IN COMPUTATIONS
XIII SLIDE RULE
XIV SOLUTION OF TRIANGLES
SECTION II APPLICATIONS OF PLANE TRIGONOMETRY
XV SMALL ANGLES
XVI PERIODIC FUNCTIONS
XVII VECTORS AND VECTOR QUANTITIES
XVIII DISTANCE—ANGULAR MEASUREMENT—SURVEYING
XIX NAVIGATION
XX MISCELLANEOUS PROBLEMS
XXI POLAR COORDINATES – COMPLEX NUMBERS—DEMOIVRE’S THEOREM—SERIES
SECTION III SPHERICAL TRIGONOMETRY
XXII INTRODUCTION
XXIII SPHERICAL TRIANGLES
XXIV APPLICATIONS OF SPHERICAL TRIGONOMETRY
APPENDIX
Customary definitions of the trigonometric functions of an acute angle
Formulas used in the text
Tables
ANSWERS TO ODD-NUMBERED PROBLEMS
INDEX
SECTION I
PLANE TRIGONOMETRY
INTRODUCTION
1. What does the word trigonometry mean?
The word trigonometry is derived from two Greek words, trigonon, “triangle,” and metria,
“measure.”
2. With what does trigonometry deal?
Trigonometry deals with triangles and other investigations involving angles.
3. How does trigonometry differ from geometry?
Geometry indicates relationships, and trigonometry supplies exact formulas for the solutions of
lengths of lines.
Geometry indicates relationships, and trigonometry provides exact formulas for calculating
unknown parts of a triangle.
4. For whom is trigonometry especially important?
Trigonometry is especially important for the navigator, astronomer, physicist, engineer, and
technical man.
5. What is analytical trigonometry?
Trigonometry that deals with proof and use of algebraic relations among the trigonometric
functions (relationships) of the same or related angles is called analytical trigonometry.
CHAPTER I
ANGLES AND THEIR MEASUREMENT
6. What is an angle?
Whenever two lines intersect at a point, the opening between the lines is called an angle. The
intersection of two streets and the gable of a roof are practical examples of angles.
7. What is the usual symbol for an angle?
The symbol ∠ in front of a designation means “angle.”
8. How are angles usually designated?
Angles are usually designated
a. by three letters with the middle letter always at the point of intersection, as ∠ABC or ∠CBA.
QUES. 8.
b. by a letter between the two lines of the angle.
c. by a letter between the two lines and an arrow between the lines to indicate the direction in which
the angle is measured.
9. What are the parts of an angle?
An angle may be considered to have an initial line, a terminal line, and a vertex. B = vertex, BC =
initial line, and BA = terminal line.
QUES. 9.
10. What are the units in which angles are measured?
The units in which angles are generally measured are a. degrees, minutes, and seconds. ° = degrees,
′ = minutes, ″ = seconds.
b. radians.
c. mils.
11. What is meant by an angle of 1 degree?
If the circumference of a circle is divided into 360 equal parts and lines are drawn from the center
of the circle to the points of division, 360 separate small angles will be formed each of which is called
an angle of 1 deg. Ninety of these small angles side by side form one right angle (90 deg.). Each of the
360 equal parts of the circumference is called an arc of 1 deg.
QUES. 11.
12. What is meant by an angle of 1 minute?
If an angle of 1 deg. is itself divided into 60 equal smaller angles, then each resulting smaller angle
is called an angle of 1 minute.
Each arc division is called an arc of 1 minute. The word minute in this sense comes from the term
minute degree.
13. What is meant by an angle of 1 second?
If an angle of 1 minute is itself divided into 60 equal still smaller angles, then each resulting still
smaller angle is called an angle of 1 second. It is obvious that an angle of 1 second is comparatively a
Each arc division is called an arc of 1 second.
very small angle.
The word second in this sense comes from the term second-minute degree.
14. What is meant by an angle of 1 radian?
If the length of an arc drawn from the vertex of the angle and between the two sides is equal to the
radius of
QUES. 14.
that arc, the angle is called an angle of 1 radian.
15. What is meant by an angle of 1 mil?
If the circumference of a circle be divided into 6,400 equal divisions and lines drawn from the
center of the circle to the points of division, 6,400 separate very small angles are formed, each of
which is called an angle of 1 mil. One thousand six hundred of these very small angles side by side
total one right angle (90 deg.).
16. Who originated the degree unit of angle?
The unit was originated by the Babylonian astronomers for convenience in their work. There is a
possible connection betweeen 360 deg. in a circle and 360, the approximate number of days in a year.
17. Where is the degree unit of angle measurement used most?
The degree unit of angle is generally used in elementary mathematics and in most practical work.
18. Why is the degree unit sometimes not used in calculus and dynamics?
The degree unit is sometimes not used in calculus and dynamics because here it sometimes makes
certain formulas more complicated than necessary.
19. Why is the radian sometimes referred to as the natural unit of angle?
The radian is referred to as the natural unit of angle because there is no arbitrary number in its
definition, as 360.
20. What is the radian measure of an angle known as?
The radian measure of an angle is known as circular measure.
21. Where is the mil unit of angle used?
The mil unit of angle is generally used in the artillery service where it is the standard unit of angle.
It is also adapted to the metric system, since it is the angle subtended by 1 meter (m.) at a range of 1
kilometer (km.).
CHAPTER II
RELATIONSHIPS OF ANGULAR MEASUREMENTS
22. What is meant by measuring a quantity?
In measuring a quantity we find its ratio to another quantity of the same kind chosen as a unit.
Examples
a. In measuring the number of degrees of an angle, we simply find the ratio of the given angle to
that of an angle of 1 deg.
b. In measuring the number of radians of an angle, we find the ratio of the length of arc of the given
angle to the radius of the arc.
c. In measuring the number of mils in an angle, we find the ratio of the given angle to that of an
angle of 1 mil.
23. What is the chief characteristic of a radian?
A radian is a constant angle, the same in all circles, because the ratio of the circumference of a
circle to its radius is constant.
Here r1, r2, and r3 are radii for different circles.
24. How is an angle measured or expressed in radians?
An angle is measured or expressed in radians by finding the ratio of the length of arc subtending the
angle to the radius of the arc.
QUES. 24.
Example
25. What angle is formed by an arc 50 inches in length with a 10-inch radius?
26. What is the radius of a circle in which an arc of 10 inches subtends an angle of 2 radians?
27. What length of arc subtends an angle of
radian if the radius of the circle is 32 feet?
28. How many degrees are there in 1 radian?
29. What part of a radian is 1 degree?
30. What is the procedure for changing from (a) degrees to radians and (b) radians to degrees?
To change
a. from degrees to radians, multiply by
or 0.017453.
b. from radians to degrees, multiply by
, or 57.296.
The approximations 0.0175 and 57.3 are sufficiently accurate for most purposes. When a specific
angular unit is not indicated, it is customary to assume that the angle is measured in radians.
31. How many degrees are there in an angle of 9 radians?
32. How many degrees are there in an angle of
radians?
33. How many radians are there in an angle of 64°42′30″?
An angle in radians is frequently expressed as a multiple of π.
34. How many radians are there in an angle of 31°46′48″?
35. May an angle contain more than 360 degrees?
In trigonometry we may deal with angles of any magnitude whatever by merely introducing the idea
of revolution.
QUES. 35.
Example
Starting from an initial position OA, revolve about O to OA1 OA2, OA3 OA4, etc., generating angles
of any number of degrees. One revolution will give 360 deg.,
revolutions, 3.5 · 360 = 1,260 deg.,
etc.
36. How is an angle read?
In an angle it is advisable that the initial line be read first, as ∠AOA1 in the above figure.
37. How are opposites indicated algebraically?
Opposites are indicated algebraically by the signs plus ( + ) and minus ( – ). If distances to the right
are assumed positive (+), then distances to the left are negative ( – ).
38. What is meant by a negative angle?
If we assume that revolution counterclockwise is the positive direction (+), then clockwise
revolution will be the negative direction ( – ), giving a negative angle of any magnitude.
and B are
and D are negative.
and ( – D + C) are the results of algebraic addition.
positive.
QUES. 38.
39. What is the relation of a mil to a radian?
For practical purposes, 1,018.6 is rounded off to 1,000, and we may then say that a mil is
radius. Therefore, 1,000 mils = 1 radian (approximately).
of the
40. What error is introduced when 1,018.6 is rounded off to 1,000?
When 1,018.6 is rounded off to 1,000, there is an error of
approximately.
41. What is the degree equivalent of a mil?
or 2 per cent,
42. What is the relation between the arc subtended, the radius, and the angle in mils?
The standard relation is
radians.
But 1 radian = 1,000 mils (approximately). Therefore,
mils, the relation between
arc, radius, and angle in mils.
Example
If a 4-yd. target is 2,000 yd. away from a gun, it subtends an angle
mils
at the gun. (For small angles the chord is taken equal to the arc subtended without too great an error.)
43. How can radian measure be useful in finding the area of a circular sector?
QUES. 43.
In a circle, areas of sectors are proportional to the central angles of the sectors.
Or the area of a circular sector is one-half the radius squared times the central angle in radians.
QUES. 44.
44. What is the length of the arc subtended by a central angle of 60 degrees in a circle of 18 inches radius, and what is the area
of the circular sector formed by the angle?
45. How is radian measure useful in considering the motion of a particle on a circle?
If a point moves from B to C in the above figure, the distance traveled s = r · A (where A is in
radians). This simple relationship leads to the simple relationship between “angular” velocity of a
rotating body and the “linear” velocity of any point on it.
If a wheel turns about its axle so that a spoke describes an angle of 6 radians/sec, then the wheel has
an angular velocity of 6 radians/sec, and a point 3 ft. from the center of the wheel moves through an
arc of s = r · A =3·6 = 18 ft. each second, or it has a linear velocity of 18 ft./sec.
In general, if a wheel has an angular velocity of A radians/sec., a point on the wheel at r distance
from the center has υ = r · A ft./sec. = linear velocity.
46. What is the procedure in adding or subtracting angles algebraically?
A. Addition
a. Place the degrees, minutes, and seconds of each angle in the appropriate column.
b. Add each column separately.
c. Redistribute excesses.
Example
B. Subtraction
Write the minuend, if necessary, in a form that will permit ready subtraction in the minute and
second columns.
Example
Write the minuend in the equivalent form 82°77′ 81″ and then subtract.
47. What is the procedure in multiplying an angle by a constant?
Multiply each unit by the constant and then reduce the result.
Example
48. What is the procedure in dividing an angle by a constant?
a. Divide the degrees by the constant.
b. Reduce any remainder of the degrees to minutes and add it to the minutes given.
c. Divide the minutes by the constant.
d. Reduce any remainder of the minutes to seconds and add it to the seconds given.
e. Divide the number of seconds by the constant.
Example
Divide 47°28′15″ by 5.
PROBLEMS
1. In the figure
a. Locate the vertex of angle A.
b. Locate the initial line of A, (A + B), B, C, (B + C), D, –E, –F, (–F + G).
c. Locate the terminal line of each of these angles.
PROB. 1.
2. Draw an angle of (a) 1,776°, (b) –116°, (c) –608°.
3. Express in degrees the angles here given in radians:
4. Express in radians, in terms of π the following angles:
5. What is the radius of the circle in which an arc of 28 in. subtends an angle of 3.6 radians?
6. In a circle with a 30-in. diameter, what angle is formed by an arc 48 in. long?
7. What length of arc subtends an angle of
radian if the radius of the circle is 44 ft.?
8. How many degrees are there in (a) 4 radians, (b) 8 radians, (c) 11 radians, (d)
radians?
9. How many degrees are there in (a) 100 mils, (b) 131 mils, (c) 64 mils?
10. How many radians are there in an angle of (a) 86°31′48″, (b) 829°16′7″, (c) –31°28′59″?
11. Through how many radians does the hour hand of a clock turn in 20 min.?
12. Through how many radians does the minute hand of a clock turn in 40 min.?
13. How many mils are there in (a) 4°18′16 ″, (b) 0.34 radian?
14. What is the length of arc subtended by an angle of 32 deg. in a 40-in.-diameter circle? What is the area of the circular sector
formed by this angle?
15. Through how many radians will a 3-ft.-diameter hind wheel of a cart turn while its 2-ft.-diameter front wheel turns 660 deg.?
16. Express the angular velocity of a wheel in radians per second if it turns 120 revolutions per minute (r.p.m.).
17. If a 4-ft. wheel turns about its axle so that a spoke describes an angle of 20π radians/sec, what is the linear velocity of a point
on the rim of the wheel?
18. What is the angular velocity of a fan in radians per second if it makes 1,200 r.p.m.?
19. What distance will the outer end of the minute hand of a clock move in 35 min. if the hand is 6 in. long?
20. What is the distance between successive graduations on a graduated quadrant if the radius of the quadrant is 3 ft. and the
graduations are 5 min. apart?
21. What must be the radius of a graduated quadrant if the distance between graduations that are 5 min. apart is to be
in.?
22. What is the area of the circular sector formed by an angle of 1 radian in a circle with a 20-in. diameter?
23. What is the length of 1 minute of arc on the equator if the radius of the earth is taken as 3,963 miles?
24. What is the linear velocity of a point on the rim of a 4-ft.-diameter flywheel if it is revolving at the rate of 4,200 r.p.m.? What is
its angular velocity?
25. What are the angular and linear velocities of a point on the rim of a 32-in.-diameter automobile tire when the car is traveling 50
miles per hour (m.p.h.)?
26. If the propeller of a Douglas combat transport is 13 ft. 2 in. in diameter, what are its tip speed and angular velocity when the
engine speed is 3,000 r.p.m.?
27. How many mils are there in 30 deg., 45 deg., and 60 deg.?
28. If a central angle in a circle 12 ft. in diameter intercepts an arc 5 in. long, how many mils are there in the angle?
29. If one mil is
of the radius, approximately, what is the length of arc (or chord for small angles) subtended by 1 mil if the
radius (distance) is (a) 1,000 ft., (b) 1,000 yd., (c) 2,000 ft., (d) 2,000 yd.?
30. How wide is a building that at 3,000 yd. subtends an angle of 20 mils?
31. The angle by which a shell misses a target that is 4,000 yd. away is 10 mils. How many yards from the target is the shell?
32. Change 46 mils to degrees, minutes, and seconds.
33. Change 0.52 radian to mils.
34. Change 32 mils to radians.
35. If a building that is known to be 70 ft. high is found to subtend an angle of 14 mils, how far from the building is the observer?
36. What is the height of a tree that is found to subtend an angle of 60 mils at 400 yd.?
Add the following angles graphically:
37. 630° and 30°
38. 900° and –30°
39. –180° and 60°
40. –45° and 120°
41. –90° and –45°
42. 135° and –450°
Evaluate each of the following algebraically:
43. (47°21'56") + (18°52'28") + (39°47'58")
44. (74°38'29") - (23°46'53")
45. (32°56'24") . 6
46. (84°48'59") . 4
47. (73°54'8") ÷ 3
CHAPTER III
CARTESIAN COORDINATES – QUADRANTS
49. What is meant by Cartesian coordinates?
By Cartesian coordinates is meant a system of coordinates, in a plane, that defines the position of a
point with reference to two mutually perpendicular lines called the axes of coordinates.
Point O is called the origin.
Lines YY′ and XX′ are called the axes of coordinates.
50. What is meant by the axis of abscissas?
Usually the horizontal line XX′ is called the axis of abscissas, or x axis.
QUES. 49–51.
51. What is meant by the axis of ordinates?
The line perpendicular to the x axis is called the axis of ordinates, or the y axis. YY′ is the axis of
ordinates.
QUES. 52.
52. In what order are the four quadrants formed by the axes of coordinates, designated?
The four quadrants that are formed by the axes of coordinates are numbered from right to left, or
counterclockwise, as shown in the figure.
53. What determines the quadrant in which an angle lies?
The terminal line of an angle determines the quadrant in which it lies.
An angle between 0 and 90 deg. is said to he in the first quadrant, since its terminal line lies in
quadrant I.
An angle between 90 and 180 deg. is in quadrant II.
An angle between 180 and 270 deg. is in quadrant III.
An angle between 270 and 360 deg. is in quadrant IV.
54. What directions are considered positive and what directions negative?
Distances measured to the right of the y axis are positive ( + )
Distances measured to the left of the y axis are negative ( – )
Distances measured above the x axis are positive (+).
Distances measured below the x axis are negative ( – ).
55. How are points located in Cartesian coordinates?
Each point is located by both its abscissa and ordinate. The abscissa is given first.
QUES. 55.
The coordinates of point P1 are abscissa x and ordinate y, written (x, y).
Point P2 coordinates are (—x, y).
Point P3 coordinates are (—x, – y).
Point P4 coordinates are (x, —y).
These show a point in each quadrant.
Note that in each case the abscissa and the ordinate are taken from the axis to the point P.
56. What is meant by the distance?
OP is called the distance of point P and is read from O to P.
QUES. 56.
57. How would you locate points (3, 5), (–3, 5), (3, –5), and ( – 3, —5) in Cartesian coordinates?
Follow the assumptions for directions and lay off first the abscissa and then the ordinate for each
point. Any suitable scale may be selected for abscissas and the same or another scale chosen for
ordinates.
58. What is a quadrantal angle?
An angle whose terminal line coincides with a quadrant line, as 0 deg., 90 deg., 180 deg., 270 deg.,
360 deg., 450 deg., etc., is called a quadrantal angle.
PROBLEMS
In what quadrant does each of the following points lie?
1. (–4, 1)
2. (2, 5)
3. (6, –2)
4. (–4, –2)
5. (–5, 1)
6. (2, –3)
7. What is the distance of each of the above points?
8. On graph paper locate the above points.
What is the distance of each of the following points?
9. (0, 3)
10. (4, 0)
11. (–2, 0)
12. (0, 0)
13. (–7, 0)
14. (–5, –3)
15. Find the abscissa of a point whose distance is 6 and whose ordinate is 5.
16. What is the abscissa of a point whose ordinate is (— 3) and whose distance is 4?
17. Prove that the ratio of the ordinate to the abscissa is constant for all points on a straight line through the origin.
18. Which of the following are quadrantal angles: 540°, 360°, 280°, –450°, 480°, 90°, (180° + A), –270°, 1,080°?
CHAPTER IV
FUNDAMENTAL TRIGONOMETRIC FUNCTIONS
59. What is a function?
A function is a relationship between variables.
x2 + y2 = r2 is a function showing a relationship between x and y.
is a function showing the relationship between s and t where s is the distance a body will
fall in t sec. and g is the constant of acceleration due to gravity.
A = πr2 is a function showing the relationship between A and r where A is the area of a circle and r
is the radius.
60. What are the functional elements of an angle?
We have seen in Ques. 9 that an angle has three parts, i.e., an initial line, a terminal line, and a
vertex.
QUES. 60.
Now, an angle may be considered to have three functional elements.
a. A “distance,” which is an assumed length on the terminal line.
b. A “shadow” cast by the assumed distance upon the initial line. (This, in other words, is the
orthographic projection of the distance on the initial line or the initial line produced.)
c. A “perpendicular” from the distance to the shadow.
61. What is a trigonometric function?
A trigonometric function is an expression showing the relationship between the functional elements
of an angle. From the three elements we can obtain six fundamental relationships.
62. What are the three direct trigonometric functions?
QUES. 62.
The three direct trigonometric functions of an angle are the sine, cosine, and tangent.
These are written sin A, cos A, and tan A, where A is any angle. They express the relationships
indicated by
63. How can these relationships or functions be readily remembered?
If it is remembered that the cosine is the shadow over the distance, then it will be recalled that the
sine is the perpendicular to the shadow over the distance and the tangent is the ratio of the
perpendicular to the shadow.
64. What are the three reciprocal trigonometric functions?
The cosecant, secant, and cotangent are the reciprocal relationships of the sine, cosine, and tangent,
respectively.
These are written