Table Of ContentTRIGONOMETRY 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: 978048644227-3 (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.
C I
HAPTER
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.
