44 PRACTICAL MATHEMATICS 



. (cos 20 + * sin 20) (cos 110 + i sin 110) 

 Example. Simplify ^ 3QO + . ^ ^^ 5QO + . ^ 5QO) 



cos 130 + i sin 130 



This becomes 3 5- 



cos 80 + * sm 80 



and finally cos 50 +i sin 50 



30. The Powers of (cos + i sin 0) . It has already been shown that 

 (cos A + * sin A) (cos B + i sin B) . . . n factors 



= cos (A + B + . . . n angles) + i sin (A + B + . . . n angles) 

 If A = B = . . . 0, making all of the angles equal 

 Then (cos + i sin Q) n = cos nQ + i sin nQ 



To raise a trigonometrical complex quantity to a power, multiply 

 the angle by the power, and the product will give the angle of 

 the resulting complex quantity. 



So far this rule only applies when the power is a positive integer. 



(a) If n is negative and this rule holds 



Then (cos + i sin Q)~ n = cos ( - nQ) + i sin ( - nQ) 



= cos nQ i sin nQ 

 and this can be proved 



for (cos + t sin 0)~ n = s ^- 



(cos + i sm 0)" 



1 



cos nQ + i sin nQ 



1 cos nQ i sin nQ 



cos nQ + i sin nQ cos nQ i sin nQ 

 _ cos nQ i sin nQ 

 ~ cos 2 nQ + sin 2 w0 

 = cos nQ i sin nQ 



Hence the rule can be applied when the power is a negative 

 integer. 



(b) If the power is a fraction and this rule holds 



Then (cos + * sin 0)^ = cos - + i sin- 



9 9 



and this can be proved 



/ . . 0V . . 



for ( cos - + * sin - ) = cos q - + ^ sm q - 



v 9 9 J 9 9 



= cos + i sin 



i 



Hence (cos + i sin 0) = cos - + i sin - 



P / 0\ 



and (cos + i sin 0) 9 = (cos - + i sin - ) 



V V 



= cos - + i sin - 



9 9 



Also (cos + i sin 0)~" = cos ( - ) + i sin ( - ) can 



\ a / \ a / 



