CHAPTER VII. ON THE RELATIONS BETWEEN THE TRIGONOMETRICAL RATIOS FOR THE SAME ANGLE, 84. LET EQP be any angle traced by QP revolving from the position QE, and let a perpendicular PM be dropped on QE or EQ produced, thus Let the angle EQP be denoted by A. Then we can prove the following relations : 85. We shall now give a number of easy examples by which the student may become familiar with the formula which we havǝ just obtained. He must observe that these formulæ hold good for all magnitudes of the angle which we have represented by the letter A, that is, not only And similarly for the other formulæ. 86. If then any angle be represented by 0, we know from Art. 84, sin 0 cos * It will be observed that in working these examples we commenced by expressing the other ratios in terms of the sine and cosine, and the beginner will find this the simplest course in most cases. The double sign before the root-symbols is to be explained thus. For an assigned value of sin ◊ we shall have more than one value of 0 (Art. 70). Hence we have an ambiguity when we endeavour to find cos from the known value of sin 0. The double sign may generally be omitted in the examples which we shall hereafter give. 89. We shall now give two examples of another method of arriving at expressions for the other ratios in terms of a particular ratio. These examples should be carefully studied. (1) To express the other trigonometrical ratios in terms of the sine. Let PAM be an angle whose sine is s, a numerical quantity. Let PM be drawn perpendicular to AM. M |
