I've been looking for a answer to why we reflect the angle whenever the length "$r$" is negative.
Why does this change the sign for $r$, when adding $180$ degrees to the angle $\theta$?
It makes sense for me how the sign might change when rotating a trigonometric functions. Like adding $\pi$ to $\sin{\frac{3\pi}{2}}$ i.e $\sin{\frac{5\pi}{2}}$ will give me a positive value for $\sin(x)$
I've tried using the the definition for
$r^2 = x^2 + y^2$, but i'm a little stuck and have yet to find a answer.
Imagine that you’re in a car that’s pointed east. Now travel in reverse for a distance $r$. Your final location can be expressed in polar coordinates as $(-r,0)$. On the other hand, you can instead turn the car around so that it faces west and then drive forward the same distance. This is expressed in polar coordinates as $(r,\pi)$. Either way, you end up in the same spot.
Algebraically, this is a straightforward consequence of the identities $$-\cos\theta = \cos(\theta+\pi) \\ -\sin\theta = sin(\theta+\pi).$$