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I,
NTRIGONOMETRY FOR BEGINNERS
BY THE
REV. J. B. LOCK, M.A.
I I
FELLOW OF GONVILLE AND CAIUS COLLEGE, CAMBRIDGE
FORMERLY MASTER AT EATON
REVISED AND ENLARGED
FOR THE USE OF AMERICAN SCHOOLS
BY
JOHN A. MILLER, A.M.
ASSISTANT PROFESSOR OF MATHEMATICS, LELAND STANFORD JR. UNIVERSITY (elect) OF MECHANICS AND MATHEMATICAL
ASTRONOMY, INDIANA UNIVERSITY'
Weto fork
THE MACMILLAN COMPANY
LONDON: & CO., Ltd.
1896
AH rights reservedCoPYRifjnT, 1S96,
By the macmillan company.
Xortuooti ^Orcss
—J. S. Cushinp & Co. Berwick & Smith
Norwood Mass. U.S.A.PREFACE
""The revision of the Trigonometry for Beginners differs
work, chiefly infrom that of the original the following par-
ticulars :
matter of Chapter VII. formerly followed thatThe subject
of Chapters VIII. and IX. ; the addition formulse are proved
for angles of any magnitude, and for more than two angles ; a
Trigonometric Functions and two chapterschapter on Inverse
on Spherical Trigonometry have been added ; logarithmic and
trigonometric tables have been inserted. The rearrangement
has necessitated minor changes in almost every chapter.
Throughout the book, the question of ambiguity of solution
It is believed that the clear,has received careful attention.
simple presentation which characterized the original work has
been retained.
It has been the endeavor to make definitions that need not
be unlearned later ; to give proofs, rigorous for the general
material the student willplane angle ; to present as much as
master in a first course ; and to present such material as will
serve him in his later studies. The proofs of many proposi-best
formulated,tions are left as exercises for the student. These are
and placed in the body of the text.
are, for theThe lists of examples in the plane trigonometry
most part, those of the original work. Some of the exercises
in spherical trigonometry are selected from other texts.
chapters andThose desiring a shorter course may omit the
the articles marked with an asterisk.
I acknowledge my indebtedness to Dr. Frank L. Sevenoak,
friends whowho kindly permitted the use of his tables, and to
aided me by suggestions.
JOHN A. MILLER.
June, 18D6.
V
183964CONTENTS
CHAPTER PAGE
I. Definitions. The Figures ok Trigonometry 1
II. Measurement of Angles...... 5
III. The Trigonometrical Ratios 12
IV. The Acute Angle 14
V. The Ratios of Certain Angles 19
VI. Practical Applications 21
VII. Use of Signs + and — , Definitions. The Trigono
metric Ratios 28
VIII. On the Relations between the TRKiOxoAiETRic Ratios 45
IX. The Solution of Trigonometrical Equations 50
X. On the Trigonometrical Ratios of Two or Mori
Angles 52
Angles 63XT. Multiple Angles, Sub-multiple
XII. Inverse Trigonometric Functions.... 68
the Sides and Angles of a TriXIII. Relations between
72angle
XIV. Logarithms 81
Area of a Triangle 92XV. Solution of Triangles.
XVI. Measurement of Heights and Distances io;5
XVII. ]\IlSCELLANEOUS THEOREMS 108
and Angles of a SpherXVIII. Relations among the Sides
CAL Triangle 121
XIX. Solution Spherical Triangles.... 13:3of
Answers to Examples 141
Tables :
Logarithms 1of Numbers
21 of Trigonometric Functions
Natural Functions .... 56
vii

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