PLANE TRIGONOMETRY. Page 10-15. Definitions of the Trigonometrical Ratios............... 16-19. Relations of the Trigonometrical Ratios to one another... 20-24. Use of the signs + and - to indicate contrariety of position 11 25-30. Magnitudes of angles unlimited. Method of reducing the Trigonometrical Ratios of all angles to those of angles less than a right angle.......... 31-34. On the angles which correspond to given values of the 35-37. Formulæ for finding the sine and cosine of the sum and 33 130-131. Demoivre's Theorem........ 132-138. Formula for expressing the sine and cosine of the sum of any angles, or of a multiple angle, in terms of the sines and cosines of the simple angles 139–140. Formulæ for expressing the powers of the sine or 1-12. Theory of Logarithms....... 13-18. Properties of Logarithms to base 10................ 19-24. Mode of using Tables of Logarithms............... Students reading this work for the first time may confine their atten- 1-27; 35-55; Appendix 1-18; 75; 89-119; 127; 129. TRIGONOMETRY. SECTION I. THEORY OF TRIGONOMETRICAL RATIOS. Object of Trigonometry.-Methods of representing numerically the magnitudes of lines and angles. 1. In any plane triangle there are six parts to be considered, three angles, and three sides. In order to find all the rest, it is in general sufficient to know three of them, but one of the three must be a side; because with three given angles (provided their sum be equal to two right angles) we can form an infinite number of triangles, which are not equal, but only similar to one another. Geometry furnishes simple constructions for each of the cases in which we can determine a triangle by means of some of its parts; but these constructions, on account of the imperfection of the instruments employed, give only a rough and often insufficient approximation. Mathematicians have therefore sought to substitute for them numerical calculations, which always attain the required degree of ex actness. The special object of Trigonometry is to give methods for calculating all the parts of a triangle when there are sufficient data; this is what is called solving a triangle. But in its present enlarged sense, Trigonometry treats of the principles by which angular magnitudes may be estimated, and numerically connected with one another, and with other magnitudes; and shews how to perform measurements generally, by means of the relations of the sides and angles of rectilinear figures. 2. To represent the magnitudes of the sides of a triangle, or of any lines, we refer them to the common unit of length, a foot for instance; and when we represent any line by a general symbol a, we use an abbreviated mode of writing a x 1; for a in reality expresses the ratio which the length of the line bears to the assumed unit of length. |