1. ## Polar Representation

Find the polar represenatation z= -1 + i.

Solution: |z| = $\sqrt{2}$ and $\theta = \frac{3\pi}{4}$. Thus,

-1 + i = $\sqrt{2}$[ $cos\frac{3\pi}{4} + i sin\frac{3\pi}{4}$]

It's been awhile since I have taken calculus, let alone any geometry. Could someone please explain to me how to find the $\theta$ in this answer?

A walkthrough of another problem could help me:

Find the polar representation for: $(2-i)^2$

Thanks for any help!

2. Hello !
Find the polar represenatation z= -1 + i.
A complex number is :
$z=|z| e^{i \theta}=|z| (\cos \theta+i \sin \theta) \quad \quad (1)$

If $z=x+iy$, the modulus of z is $|z|=\sqrt{x^2+y^2}$
Here, $z=-1+i \implies x=-1 \text{ and } y=1 \implies |z|=\sqrt{(-1)^2+1}=\sqrt{2}$

So factor by $\sqrt{2}$ :

$z=-1+i=\underbrace{\sqrt{2}}_{|z|} \left(\frac{-1}{\sqrt{2}}+i \frac{1}{\sqrt{2}} \right)$

According to relation (1), $\cos \theta+i \sin \theta=\frac{-1}{\sqrt{2}}+i \frac{1}{\sqrt{2}}$

When 2 complex numbers are equal, their real parts are equal and their imaginary parts are equal.

Therefore $\cos \theta=\frac{-1}{\sqrt{2}}=- ~\frac{\sqrt{2}}{2}$ and $\sin \theta=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$