34 PRACTICAL MATHEMATICS 



When A = B sin 2A = 2 sin A cos A 



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

 cos 2A = cos 2 A sin 2 A 

 = 2 cos 2 A - 1 

 = 1-2 sin 2 A 

 2 sin A cos A 



tan 2A = 



cos 2 A sin 2 A 

 2 tan A 



1 - tan 2 A 

 Also sin 2 A = - (1 - cos 2A) 



2 



1 



2 



2 "" 2 = 



and cos 2 A = - (1 + cos 2 A) 



00/1 

 Putting A = - sin - = A/- (1 - cos 0) 



cos - = \ - 



/I 



2 = Vi 



, * cos 

 tan 



+ cos 



(l-cos0) 2 



1 - cos 2 

 = 1 - cos 



sin 

 sin 3A = sin (2A + A) = sin 2A cos A + cos 2A sin A 



= 2 sin A cos 2 A + sin A (1 - 2 sin 2 A) 

 = sin A (2(1 - sin 2 A) + 1 - 2 sin 2 A} 

 = sin A (3 - 4 sin 2 A) 



The relations for cos 3A, sin 4A, etc., can be obtained in a similar 

 manner, and they can be well left as exercises for the student. 



Example. The general term of the series for t?* sin bx is 



n xviH lj 



(a 2 + & 2 )* : sin no. where tan a = Taking a = 2, b = 1, find 



I tl CL 



the first five terms of the series. 



Then e sin bx = (a 2 + 6 2 )* x sin a + (a 2 + b 2 ) ^! sin 2a + . . . 



I? 



a; 2 x^ / 



and e zx sin x = xV5 sin a -f r^r 5 sin 2a + r^ 5 V 5 sin 3<x . . . 



'_ '_ 



Thus to obtain the first five terms we have to find the values of 



sin a, sin 2a, sin 3a, sin 4a, and sin 5a, knowing that tan a = -. 



m 



