Prove that $\left(r^{2} \cos 2 \theta, r^{2} \sin 2 \theta\right)$ is on upper half-space

Question

Prove $\left(r^{2} \cos 2 \theta, r^{2} \sin 2 \theta\right)$ is on the upper half space.

( assuming $x=r \cos \theta, y=r \sin \theta$ for $r>0,0<\theta<\pi / 2$)

Is there rigorous proof without substituing the endpoints?

Answer 1

To show that your values are in the upper half space, it suffices to show that its $y$ coordinate is positive, i.e. that $r^2\sin 2\theta$ is positive for $0<\theta<\frac{\pi}{2}$. Well, it is clear that $r^2 > 0$ and also recall that $\sin(x) \geq 0$ for $ 0\leq x\leq\pi$. Taking $x = 2\theta$, we get the desired result.