Question

Find the quotient of 3(cos135° + isin135°) and 9(cos45° + isin45°).

Answer 1

Quotient is i/3.

Explanation:

Given:

Complex numbers are 3( cos 135° + i sin 135° ) and 9( cos 45° + i sin 45° )

To find: Quotient of the given complex number.

Consider,

`(3(cos\,135+i\:sin\,135))/(9(cos\,45+i\:sin\,45))`

`(3)/(9)*(cos\,135+i\:sin\,135)/(cos\,45+i\:sin\,45)`

`=(1)/(3)*(cos\,135+i\:sin\,135)/(cos\,45+i\:sin\,45)*(cos\,45-i\:sin\,45)/(cos\,45-\:sin\,45))`

`=(1)/(3)*((cos\,135+i\:sin\,135)(cos\,45-i\:sin\,45))/((cos\,45-i\:sin\,45)(cos\,45+\:sin\,45))`

`=(1)/(3)*((cos\,135\:cos\,45+sin\,135\:sin\,45+i(sin\,135\:cos\,45-cos\,135\:\:sin\,45))/(cos^2\,45-(i\:sin45)^2)`

using, cos A cos B + sin A sin B = cos( A - B ) and sin A cos B - cos A sin B = sin( A - B )

`=(1)/(3)*(cos\,(135-45)+i\:sin\,(135-45))/(cos^2\,45-(-1)sin^2\,45)`

`=(1)/(3)*(cos\,90+i\:sin\,90)/(cos^2\,45+sin^2\,45)`

`=(1)/(3)*(0+i)/(1)`

`=(i)/(3)`

Therefore, Quotient is i/3.

Answer 2

`(1)/(3√(2) ) (1+i)`

Explanation:

Given are two complex numbers as

z1 = 3 (cos 135+isin 135)

z2 = 9(cos 45+isin45)

To find quotient

We can use Demoivre theorem for products and quotients here

`(z1)/(z2) =(3(cos135+isin135))/(9(cos45+isin45)) \\=(1)/(3) (cos135-90+isin 135-90)\\=(1)/(3) (cos45 +isin 45)\\=(1)/(3√(2) ) (1+i)`