MULTIPLE ANGLES 35 



112 



If tan a = -, then sin a = 7= and cos a = 7= 



V5 V5 



124 413 



sin 2a - 2 r= . 7= = -, cos 2a - = - 



A/5 V5 5 555 



4281 11 3241 2 



sin 3a - - 73 + - 7= = 7=, cos 3a = --- r= ---- T= -= 7=. 

 5 A/5 5 A/5 5A/5 5 A/5 5 A/5 5A/5 



4 3 24 9 16 7 

 sin 4a = 2 - - = , cos 4a = - = - 



5 5 25 25 25 25 



24 2 7 1 41 



sin 5a = 7= -- . 7= = - 7-=., 



25 V5 25 A/5 25 A/5 



7 _2__24 J_ 38 



~ 25 A/5 25 ' A/5 ~ 25A/5 



._ 1 , # 2 4 ar 11 .r 4 24 



11 41 



** + ^ 



21. Now sin (A + B) = sin A cos B + cos A sin B . . . (1) 



sin (A B) = sin A cos B cos A sin B . . . (2) 



cos (A + B) = cos A cos B sin A sin B . . . (3) 



cos (A B) = cos A cos B + sin A sin B . . . (4) 



Adding (1) and (2) 



sin (A + B) + sin (A - B) = 2 sin A cos B . . . (5) 

 Subtracting (2) from (1) 



sin (A + B) - sin (A - B) = 2 cos A sin B . . . (6) 

 Adding (3) and (4) 



os (A + B) + cos (A - B) = 2 cos A cos B . . . (7) 

 Subtracting (3) from (4) 



cos (A - B) - cos (A + B) = 2 sin A sin B . . . (8) 

 In these relations A is taken as being greater than B. 

 The relations (5), (6), (7), and (8) can be used in two different 

 ways : 



(a) To express the sums or differences of sines or cosines as the 



products of sines and cosines. 



(b) To express the products of sines and cosines as the sums 



or differences of sines or cosines. 

 (a) Comparing sin (x + h) sin x with relation (6). 

 Then A + B = x + h and A B = x 

 Ji It 



Hence A = x + ^ and B = - 



'- & 



and sin (x + h) sin x = 2 cos \x + -) sin - 



\ ^/ ^ 



