When the angle A is 0°, MP is zero, and when A is 90°, MP is equal to OP; and as A continuously increases from 0° to 90°, MP increases continuously from zero to OP; also OP is always equal to Ο.Χ. Therefore, when A=0°, the fraction MP is equal to that is OP' 0; when A = 90°, the fraction is equal to MP A continuously increases from 0° to 90°, the numerator of the fraction MP ОР continuously increases from zero to OP, while the denominator is MP unchanged, and therefore the fraction which is sin A, increases continuously from 0 to 1, and is positive. OP' As A increases from 0° to 90°, MP increases from zero to OP, and is positive. Therefore sin A increases from 0 to 1, and is positive. As A increases from 90° to 180°, MP decreases from OP to zero, and is positive. Therefore sin A decreases from 1 to 0, and is positive. As A increases from 180° to 270°, MP increases numerically from 0 to OP, and is negative; hence sin A increases numerically from 0 to 1, and is negative. As A increases from 270° to 360°, MP decreases from OP to zero, and is negative. Therefore sin A decreases numerically from 1 to 0, and is negative. COROLLARY. Therefore we may conclude that sin A is never greater than 1; that cos A is never greater than 1. That the numerical value of sec A or of cosec A is never less than 1. For Trace the changes in sign and magnitude as A increases from 0° to 360° of Let XOPA, any angle. (The figures are those of Art. 25.) The following relations are evident from the definitions: 68. The following is a List of Formula with which the student must make himself familiar: 69. By means of these formulæ we are able to transform a given trigonometrical expression into a great variety of equivalent expressions. Prove that tan A+ cot A sec A cosec A. EXAMPLE. = 70. It is sometimes convenient to write a given expression in terms of the sine only, or in terms of the cosine only. EXAMPLE I. Prove that sin1 0 + 2 sin2 0 cos2 0 = 1 − cost 0. |