Write each of the given numbers in the polar form reiθ , −π<θ≤π −π<θ≤π .

Question

Write each of the given numbers in the polar form re^iθ , −π<θ≤π −π<θ≤π.

 

1). (2-2i)/(-sqrt(3) + i)

 r = ? and theta =?

 

2).  (2i)/(3e⁽⁸⁺ⁱ⁾⁾

r = ? and theta = ?

Answer 1

1.

(2-2i)/(-√3 + i)=((2-2i)(-√3 - i))/((-√3 + i)(-√3 - i))

=(-2(1+√3)+ i2(√3-1))/4=-(1+√3)/2+i(√3-1)/2

r=√[((1+√3)/2)²+((√3-1)/2)²]=√2

θ=arctan((√3-1)/(√3+1))=11π/12

(2-2i)/(-√3 + i)=√2e^11π/12

 

2)

(2i)/(3e⁽⁸⁺ⁱ⁾ =(2/3e⁸)ie^-i/eⁱ)(e^-i)=(2/3e⁸)ie^-i

=(2/3e8)i[(cos(1)-isin(1)]=(2/3e8)[(sin(1)+icos(1)]

r=(2/3e⁸)

θ=π/4. 

So

 (2i)/(3e⁽⁸⁺ⁱ⁾⁾=(2/3e⁸)e^π/4=(2/3)e^8+π/4.