DE MOIVRE'S THEOREM 43 



behaviour of the portion (cos 6+ i sin 6), for r, being a positive 



number, is quite easily dealt with. 



(cos A +i sin A)(cos B +i sin B) =cos A cos B 4-i 2 sin A sin B 



+t(sin A cosB +cos A sin B) 

 = (cos A cos B sin A sin B) 

 + i(sin A cos B +cosA sin B) 

 - cos (A + B) + i sin (A + B) 



Thus the product of two trigonometrical complex quantities 

 gives a trigonometrical complex quantity, the angle of which is 

 the sum of the two angles in the factors. This can be extended 

 to the product of any number of factors. 

 For (cos A 4- i sin A) (cos B + i sin B)(cos C + i sin C) 

 = {cos (A + B) + i sin (A + B) }(cos C + i sin C) 

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

 and in general (cos A + i sin A) (cos B + i sin B) . . . n factors 



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

 Division can be performed by representing the process by a 

 fraction and then simplifying that fraction. 

 To divide (cos A + i sin A) by (cos B + i sin B) we must simplify 



cos A + i sin A 



the fraction rr ^-. ^, and this can be done by multiplying 



cos B + i sin B 



numerator and denominator by cos B i sin B. 



cos A 4- i sin A cos A + i sin A cos B i sin B 



1 lien. = x ---- i 



cos B + i sin B cos B + i sin B cos B i sin B 



cos A cos B i 2 sin A sin B + i (sin A cos B cos A sin B } 



cos 2 B - i 2 sin 2 B 

 (cos A cos B + sin A sin B) + i(sin A cos B cos A sin B) 



cos 2 B + sin 2 B 

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



Thus division or the simplification of a fraction gives a trigono- 

 metrical complex quantity whose angle is the angle in the numerator 

 diminished by the angle in the denominator. Thus, in general, 

 if we have a fraction whose numerator is the product of factors 

 of the form (cos A.+ i sin A), and whose denominator is the 

 product of factors of the form (cos a + i sin a), the fraction reduces 

 down to (cos 6 + i sin 0), where is the sum of the angles in the 

 numerator diminished by the sum of the angles in the denominator. 



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

 (cos a + i sin a) (cos } + i sin (3) . . . m factors 



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

 ~~ cos (a + (3 + . . . m angles) + i sin (a + (3 + . . . m angles) 

 = cos {(A + B . . . n angles) (a + (3 . . . m angles) } 



+ i sin {(A + B . . . n angles) (a + (i . . . m angles) } 

 = cos + i sin 



