56 88. TRIGONOMETRY. We proceed to show that the Trigonometrical Ratios of an angle vary in Sign according to the Quadrant in which the revolving line of the angle happens to be. From the definition we have, with the usual letters, I. When OP is in the first Quadrant (Fig. 1.). MP is positive because from M to P is upwards (Rule 11. p. 55.) OM is positive because from 0 to M is towards the right, OP is positive. (Rule 1.). (Rule III.). Hence, if A be any angle of the first Quadrant, Globe 8vo. Price 4s. 6d. Key, 8s. 6d. A TREATISE ON ELEMENTARY TRIGONOMETRY BY REV. J. B. LOCK, M.A. FELLOW OF GONVILLE AND CAIUS COLLEGE, CAMBRIDGE, FORMERLY MASTER AT ETON. STEREOTYPED EDITION London MACMILLAN AND CO., LIMITED NEW YORK: THE MACMILLAN COMPANY All rights reserved CONTENTS Chapter I. On Measurement; II. On Incommensurable Quantities; III. On the Relation between the Circumference of a Circle and its Diameter; IV. On the Measurement of Angles; V. The Trigonometrical Ratios; VI. On the Ratios of Certain Angles; VII. On the Trigonometrical Ratios of the same Angle; VIII. On the Use of the Signs and; IX. On the Use of and in Trigonometry; X. On Angles unlimited in Magnitude, I.; XI. On the Ratios of Two Angles; XII. On the Ratios of Multiple Angles; XIII. On Angles unlimited in Magnitude, II.; XIV. On Logarithms; XV. On the Use of Mathematical Tables; XVI. On the Relations between the Sides and Angles of a Triangle; XVII. On the Solution of Triangles; XVIII. On the Measurement of Heights and Distances; XIX. On Triangles and Circles; XX. On the Area of the Circle, the Construction of Trigonometrical Tables, etc., Appendix, Examples for Exercise, Examination Papers, Answers to Examples. and and SIDES AND ANGLES OF A TRIANGLE. 241. Let s stand for half the sum of a, b, c, so that (a+b+c) = 28. 199 Then, (b+ca) = (b + c + a−2a)=(28-2a)=2(8-a), (c + a − b) = (c + a + b − 2b) = (28 — 2b) = 2 (8 — b), (a + b −c) = (a + b + c − 2c) = (28 — 2c) = 2 (8 — c). 242. and that VI. To prove that sin COS H bc when s stands for half the sum of the sides a, b, c. ON TRIANGLES AND CIRCLES. 231 274. To find the Radius of the Circumscribing Circle. Fig. 1. B R A Let a circle AA'CB be described about the triangle ABC. Let R stand for its radius. Let O be its centre. Join BO, and produce it to cut the circumference in A'. Join A'C. Then, Fig. 1. the angles BAC, BA'C in the same segment are equal; Fig. 11. the angles BAC, BA'C are supplementary; also the angle BCA' in a semicircle is a right angle. Thus d, the value of each of these fractions, is the diameter of the circumscribing circle. Globe 8vo. 4s. 6d. ELEMENTARY TRIGONOMETRY BY H. S. HALL, M.A. FORMERLY FELLOW OF CHRIST'S COLLEGE, CAMBRIDGE AND S. R. KNIGHT, B.A., M.B., CH.В. FORMERLY SCHOLAR OF TRINITY COLLEGE, CAMBRIDGE London MACMILLAN AND CO., LIMITED NEW YORK: THE MACMILLAN COMPANY The distinctive features of the book are (1) The subdivision of the chapters into short sections, each followed by carefully graduated examples. (2) The special prominence given to easy Identities and Equations, as soon as the fundamental properties of the Trigonometrical Ratios have been discussed. (3) The postponement of radian or circular measure, a subject which has usually been placed so as to intimidate the beginner at the very outset. (4) The model solutions of logarithmic questions, and of problems in heights and distances. A KEY, for the use of Teachers. Crown 8vo. 8s. 6d. |