1. ## Polar Representation

Find the polar represenatation z= -1 + i.

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

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

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

A walkthrough of another problem could help me:

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

Thanks for any help!

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

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

So factor by $\displaystyle \sqrt{2}$ :

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

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

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

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