TRIGONOMETRY AND TABLES BY G. A. WENTWORTH AUTHOR OF A SERIES OF TEXT-BOOKS IN MATHEMATICS SECOND REVISED EDITION BOSTON, U.S.A. GINN & COMPANY, PUBLISHERS The Athenæum Press (Second Revised Edition) Analytic Geometry Logarithms and Metric Measures Geometrical Exercises Syllabus of Geometry Examination Manual in Geometry (Wentworth and Hill) Exercise Manual in Geometry (Wentworth and Hill) Plane Trigonometry (Second Revised Edition) Plane Trigonometry and Tables (Second Revised Edition) Plane and Spherical Trigonometry (Second Revised Edition) Plane and Spherical Trigonometry, with Tables (Second Revised Edition) Plane Trigonometry and Surveying, with Tables (Second Revised Edition) Surveying and Tables (Second Revised Edition) Plane and Spherical Trigonometry and Surveying, with Tables (Second Revised Edition) Plane and Spherical Trigonometry, Surveying, and Seven Tables (Wentworth and Hill) COPYRIGHT, 1882, 1895, 1902, 1903, BY G. A. WENTWORTH ALL RIGHTS RESERVED PREFACE IN preparing this work the aim has been to furnish just so much of Trigonometry as is actually taught in our best schools and colleges. Consequently, all investigations that are important only for the special student have been omitted, except the development of functions in series. The principles have been unfolded with the utmost brevity consistent with simplicity and clearness, and interesting problems have been selected with a view to awaken a real love for the study. Much time and labor have been spent in devising the simplest proofs for the propositions, and in exhibiting the best methods of arranging the logarithmic work. The author acknowledges his obligation to G. A. Hill, A.M., of Cambridge, Mass., to Dr. F. N. Cole, of New York, N.Y., to Professor S. F. Norris, of Baltimore, Md., and to Professor B. F. Yanney, of Alliance, Ohio. EXETER, N.H., G. A. WENTWORTH. [The numbers refer to the pages.] CHAPTER I. TRIGONOMETRIC FUNCTIONS OF ACUTE ANGLES: Angular measure, 1; trigonometric functions, 3; representation of the functions by lines, 7; changes in the functions as the angle changes, 10; functions of complementary angles, 11; relations of the functions of an angle, 13; formulas for finding all the other functions of an angle when one function of the angle is given, 15; functions of 45°, 17°; functions of 30° and 60°, 18. Given parts of a right triangle, 20. 20; Case I, when an acute angle and the hypotenuse are given, 20; Case II, when an acute angle and the opposite leg are given, 21; Case III, when an acute angle and an adjacent leg are given, 21; Case IV, when the hypotenuse and a leg are given, 22; Case V, when the two legs are given, 22; general method of solving the right triangle, 23; solutions by logarithms, 25; area of the right triangle, Definition of goniometry, 36; positive and negative quantities, 36; co-ordinates of a point in a plane, 37; angles of any magnitude, 38; functions of any angle, 40; functions of a variable angle, 42; func- tions of angles larger than 360°, 44; extension of formulas for acute angles to angles of any magnitude, 44; reduction of the functions of all angles to the functions of angles in the first quadrant, 47; func- tions of angles that differ by 90°, 50; functions of a negative angle, 51; functions of the sum of two angles, 53; functions of the differ- ence of two angles, 56; functions of twice an angle, 58; functions of half an angle, 58; sums and differences of functions, 59; anti- |