Show $T(r, \theta) = (r \cos \theta , r \sin \theta)$ is injective

Question

So I need to show $r = r'$ and $\theta = \theta '$ using: $$r \cos \theta = r' \cos \theta ' $$ and $$r \sin \theta = r' \sin \theta '$$ I don't know how to solve this system of equations because we have 4 unknown variables with 2 equations and we need to show $r = r'$ and $\theta = \theta '$.

Answer 1

I'm assuming that domain of $T$ is $(0,\infty)\times[0,2\pi)$.

Square both equations and add them to obtain: $$r^2=r'^2$$ so $r=r'$.

Now the angles $\theta$ and $\theta'$ have the same $\sin$ and $\cos$. Hence, they are the same angle.