Finding polar coordinates for this triangular region

Question

I want to find polar coordinates for the red triangle; here is a picture.

So $ \theta \in [\pi/4,\pi/2]$ ,but I am not sure what $r$ might be, this is what I did:

Using the angle marked in green from the picture and $r$ as the hypotonuse

$cos\theta = 1/r$

$ r = 1/cos\theta$ and $ r \in [0,1/cos\theta]$

Answer 1

For a fixed angle $\theta$, the points of the region corresponding to the angle $\theta$ are bordered from the point $(0,0)$ to the point of the line $y=1$. In the polar coordinates the point $(0,0)$ means $r=0$ and the line $y=1$ has the equation $r\sin\theta=1$ or equivalently $r=\frac{1}{\sin\theta}$.

So $r\in\left[0,\frac{1}{sin\theta}\right]$.