PLANE AND SPHERICAL TRIGONOMETRY. BY G. A. WENTWORTH, A.M., FROFE8SOR OP MATHEMATICS IN PHILLIPS EXETER ACADEMY. BOSTON, U.S.A.: Entered, according to Act of Congress, in the year 1883, by G. A. WENTWORTII, in the Office of the Librarian of Congress, at Washington. PREFACE. N preparing this work the aim has been to furnish just so much -L of Trigonometry as is actually taught in our best schools and colleges. Consequently, development of functions in series and all other investigations that are important only for the special studeut have been omitted. 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 object of the work on Surveying and Navigation is to present these subjects in a clear and intelligible way, according to the best methods in actual use; and also to present them in so small a compass that students in general may find the time to acquire a competent knowledge of these very interesting and important studies. The author is under particular obligation for assistance to Q. A. Hill, A M., of Cambridge, Mass., and to Prof. James L. Patterson, of Lawrenceville, N.J. G. A. WENTWORTH. Phillips Exeter Academt, CONTENTS. CHAPTER I. Functions Of Acute Angles: Definitions, 1; representation of functions by lines, 7; changes in the functions as the angle changes, 9; functions of complementary angles, 10; relations of the functions of an angle, 11; formulas for finding all the other functions of an angle when one function of the angle is given, 13; functions of 45°, 30°, 60°, 15. CHAPTER II. The Right Triangle: Solution: Case I., when an acute angle and the hypotenuse are given, 16; Case II., when an acute angle and the opposite leg are given, 17; Case III., when an acute angle and the adjacent leg are given, 17; Case IV., when the hypotenuse and a leg are given, 18; Case V., when the two legs are given, 18; general method of solving the right triangle, 19; area of the right triangle, 20; the isosceles triangle, 24; the regular polygon, 26. Definition of Goniometry, 28; angles of any magnitude, 28; gen- eral definitions of the functions of angles, 29; algebraic signs of the functions, 31; functions of a variable angle, 32; functions of angles larger than 360°, 34; formulas for acute angles extended to all angles, 35; reduction of the functions of all angles to the functions of angles in the first quadrant, 38; functions of angles that differ by 90°, 40; functions of a negative angle, 41; functions of the sum of two angles, 43; functions of the difference of two angles, 45; functions of twice an angle, 47; functions of half an angle, 47; sums and differences of |