Inequality for $\frac{\sin x}{\sin y}$

Question

We know that $$\sqrt{\sin x \sin y}\leq \sin\left(\frac{x+y}{2}\right)$$

Is there an useful inequality for $\dfrac{\sin x}{\sin y}$ like this?

$$\frac{\sin x}{\sin y}\leq f(x,y)$$

Answer 1

Not really, unless you count $f(x,y) = \csc y$.

$\displaystyle \frac{\sin x}{\sin y}$ is unbounded, unlike $\sqrt{\sin x \sin y}$, and when $x = (2n+\frac{\pi}{2})\pi$, it equals $\csc y$.