# Thread: Proof of polar coordinates

1. ## Proof of polar coordinates

Let z e C.

0 is how I write theta below !!!

If z = r cos(0) + r sin(0) i in polar coordinates, then

z^-1 = r^-1 cos (-0) + r^-1 sin(-0) i and,

z^-1 = r^-1 cos (0) - r^-1 sin(0) i

How do I prove this? Thanks in advance!

2. Originally Posted by jzellt
Let z e C.

0 is how I write theta below !!!

If z = r cos(0) + r sin(0) i in polar coordinates, then

z^-1 = r^-1 cos (-0) + r^-1 sin(-0) i and,

z^-1 = r^-1 cos (0) - r^-1 sin(0) i

How do I prove this? Thanks in advance!
This is an application of DeMoivre's Theorem...

$\displaystyle z^n = r^n(\cos{(n\theta)} + i\sin{n\theta})$.

So $\displaystyle z^{-1} = r^{-1}[\cos{(-\theta)} + i\sin{(-\theta)}]$.

These are some well-known identities:

$\displaystyle \cos{(-\theta)} = \cos{\theta}$ and $\displaystyle \sin{(-\theta)} = -\sin{\theta}$

(Check their graphs if you have to)...

So $\displaystyle z^{-1} = r^{-1}(\cos{(-\theta)} + i\sin{-\theta}) = r^{-1}(\cos{\theta} - i\sin{\theta})$.