Question

How do you express `z = 1 + i sqrt(3)` in polar and trigonometric form, and sketch z on the complex number plane?

Answer 1

The polar form is `(2, pi/3)`. The trigonometric form is `z=2(cos(pi/3)+isin(pi/3))`

Explanation:

The polar form is `r=f(theta)`

`z=costheta+isintheta`

The modulus of `z` is

`|z|=|1+sqrt(3)i|=sqrt(1+(sqrt3)^2)=sqrt4=2`

`r=z/(|z|)=(2)(1/2+sqrt(3)/2i)`

So,

`costheta=1/2` and `sintheta=sqrt(3)/2`

Therefore,

`theta=pi/3` `mod 2pi`

So,

The polar form is `(2,pi/3)`

The trigonometric form is `z=2(cos(pi/3)+isin(pi/3))`

graph{((x-1)^2+(y-sqrt3)^2-0.001)=0 [-1.313, 4.163, -0.42, 2.316]}