42 PRACTICAL MATHEMATICS 



The angle must be such that its sine is positive and its cosine 

 negative, an angle between 90 and 180. Thus = 180 - A. 

 Therefore - a + bi = A/a 2 + 6 2 {cos(180 - A) + ism (180 - A)} 



where tan A = -. 

 a 



Case III. When a and b are both negative. 

 Then a bi = r(cos + i sin 0) 



and r cos = a and r sin = b 



The angle must be such that its sine and cosine are both 

 negative : an angle between 180 and 270. Thus = 180 + A. 

 Therefore -a-bi = Va 2 + 6 2 {cos ( 180 + A ) + * sin (180 + A) } 



where tan A = -. 

 a 



Case IV, When a is positive and b is negative. 

 Then a bi = r(cos + i sin 0) 



and r cos = a and r sin = b. 



The angle must be such that its sine is negative and its cosine 

 positive : an angle between 270 and 360. Thus = 360 -A. 

 Therefore a - bi = Va z + ft 2 {cos (360 - A) + i sin (360 - A) } 



where tan A = -. 

 a 



Example 1. Express 7+5* in the form r(cos + i sin 0). 

 Then 7 + 5i = r(cos + i sin 0) 

 and r cos = 7, r sin =5 

 r = \/49 + 25 = V74 = 8-602 

 Since sin is + and cos is - 



Then = 180 - A, where tan A = | = 0-7143 



= 180 - 35 32' 



= 144 28' 

 and - 7 + 5i = 8-602(cos 144 28' + i sin 144 28') 



Example 2. Express 8 Hi in the form r(cos + i sin 0). 

 Then - 8 - Hi = r(cos + f sin 0) 

 and r cos = 8, r sin = 11 



r = V64 + 121 = Vl85 = 13-60 

 Since sin is and cos is 



Then = 180 + A, where tan A = ^ = 1-375 



o 



= 180 + 52 59' 

 = 232 59' 



and - 8 - Hi - 13-60(cos 232 59' + i sin 232 59') 



29. Multiplication of Trigonometrical Complex Quantities. In 

 multiplication and division we are more concerned with the 



