PRACTICAL MATHEMATICS 



Let P (Fig. 7) be a point taken on the moving arm such that 

 OP is the same length in each case. From P draw a perpendicular 

 to the fixed arm. In Cases II and III this perpendicular meets 

 the fixed arm produced. The triangle OPR will give us the 

 trigonometrical ratios of the angle in each case. 



If in each case the angle POR is made equal to a, then the 



CASE. 2. 



CASE. 



CASE.3 



CASE.A. 



o 



FIG. 7. 



right-angled triangle OPR is equal in all respects, for four distinct 

 positions of the moving arm OP. Thus the angles a, 180 a, 

 180 + a, and 360 a have the common property that their 

 trigonometrical ratios are all derived from the same right-angled 

 triangle, and therefore the sines of these angles will all be numeri- 

 cally equal. This also holds for the other trigonometrical ratios. 



