PLANE AND SPHERICAL TRIGONOMETRY BY JAMES M. TAYLOR, A.M., LL.D. PROFESSOR OF MATHEMATICS, COLGATE UNIVERSITY; AUTHOR OF GINN & COMPANY BOSTON NEW YORK. CHICAGO LONDON cop. 1963. Frz. 8986 JUL 24 1906 Harvard University, C'ft of the Publishers. TRANSFERRED TO COPYRIGHT, 1904, 1905 BY JAMES M. TAYLOR ALL RIGHTS RESERVED 55.8 The Athenæum Press PREFACE This book is designed for those who wish to master the fundamental principles of Trigonometry and its most important applications. It is adapted to the use of colleges and high schools. The proofs of formulas are simple but rigorous. The use of directed lines is consistent; the directions of such lines. in the figures are usually indicated by arrowheads, and these lines are always read from origin to end. Both trigonometric ratios and trigonometric lines are employed, but at first the ratios are used exclusively until they have become fixed in the mind and have been made familiar by use in the solution of right triangles. The distinction between identities and equations is recognized in definition, notation, and treatment. The solution of trigonometric equations is scientific and complete. The trigonometric ratios are defined in pairs as reciprocals of each other both to aid the memory and to emphasize one of the most important of their fundamental relations. The addition formulas are proved for positive or negative angles of any quadrant, and from them are deduced the other formulas concerning the functions of two or more angles. When two or more figures are used in a proof, the same phraseology always applies to each figure. In the first chapter, by means of the right triangle, the pupil is taught some of the uses of Trigonometry before he is required to master the broader ideas and relations of Analytic Trigonometry; but at the same time the emphasis is so distributed that when the general ideas are taken up they easily replace the special ones. In Chapter VIII complex number is expressed as an arithmetic multiple of a quality unit in its trigonometric type form, and the fundamental properties of such number are demonstrated. The proof of De Moivre's theorem is simple but complete, and its meaning and uses are illustrated by examples. In Spherical Trigonometry the fundamental relations of spherical angles and triangles to diedral and triedral angles are illustrated by constructions. The complete solution of the right triangle is discussed by itself, but later the formulas used are shown to be only special cases of the laws of sines and cosines for the oblique triangle. The most useful and interesting problems have been selected and special attention has been given to methods of solution and to arrangement of work. It is believed that the order of the text is the best for beginners; but, with the exception of a few articles, Chapter I or Chapter X may be omitted by those who are prepared to take up at once the general treatment in Chapter II or Chapter XI. Too much stress cannot be laid on careful and accurate construction and measurement in the first chapters. Chapters VII and VIII and the latter part of Chapter VI may be omitted by those who wish a shorter course. In writing this book the author has consulted the best authorities, both American and European. Many of the examples have been taken from these sources. The author takes this opportunity to express to many teachers and other friends his appreciation of their valuable suggestions in the course of the preparation of the book. COLGATE UNIVERSITY, May, 1905 JAMES M. TAYLOR 4. Construction of angles having given ratios 5,6. Approximate values, and changes from 0° to 90° 7-9. Trigonometric ratios of angles of triangle and of co-angles 3, 4 5-7 8,9 10, 11 11-13 14-21 40-43 26. Changes in ratios as angle changes from 0° to 360°. 27. Trigonometric ratios of 0°, 90°, 180°, 270° 28, 29. Trigonometric ratios of A and 90° + A |