CHAPTER II TRIGONOMETRIC FUNCTIONS 20. The trigonometric functions, upon which trigonometry is based, are functions of an angle. These functions are the sine, cosine, tangent, cotangent, secant, cosecant, versed sine, and coversed sine of an angle. For any angle a they are written sin a, cos a, tan a, cot α, sec a, csc a, vers a, and covers a. 21. Definitions of the trigonometric functions. Let OX and OP be the initial and terminal lines respectively of any angle α. Let P be any point on the terminal line, OP or the distance from the origin to the point P. OA or x the abscissa of the point P, and AP or y the ordinate of the point P. It should be noted that OP, OA, and AP are directed lines and hence r = OP and not PO x= OA and not AO y= AP and not PA. These are It will be observed that each of the first six functions is defined as a ratio between two line segments. the fundamental trigonometric functions, and the ratios defining them are called the trigonometric ratios. The first six functions are called trigonometric ratios. The expression trigonometric functions is more general and embraces the versed sine, coversed sine, and the trigonometric ratios. It is evident, from the definitions, that the values of the trigonometric functions are abstract numbers. 22. Values of the trigonometric functions of 30°, 45°, and 60°. A few concrete illustrations serve to show the nature of the trigonometric functions, to fix ideas, and to prepare the way for more general considerations. . Functions of 30°. Let OPF be an equilateral triangle having its sides equal to 2 units. Place the triangle with a vertex at the origin so that OX bisects the angle POF. Then, by geometry, the angle AOP 30°, the ordinate AP=1, and the abscissa OA= √3. = PROBLEM. Find the values of the trigonometric functions of 30°, as above, taking OP=1. Functions of 45°. Let OAP be Y P PROBLEM. Find the values the functions of 60°, as above, taking OP=4 a. The values of the sines and cosines of 30°, 45°, and 60° are used frequently and should be memorized. The following table may be found helpful: 23. Values of the trigonometric functions of 120°, 135°, and 150°. By using the magnitudes of the figures of Art. 22 and properly placing them with respect to the axes, the |