Solving for $a$ in $\arcsin\sqrt a = \arcsin\sqrt b - \dfrac h2$

Question

How can I solve

$$\arcsin\sqrt a = \arcsin\sqrt b - \dfrac h2$$

I would like solve for $a$ in terms of $b$ and $h$.

Answer 1

Very simple: Just take the sine of both sides and square:

$$\sqrt{a} = \sin \left( \sin^{-1} (\sqrt{b}) - h/2\right)$$

or

$$a = \sin^2 \left( \sin^{-1} (\sqrt{b}) - h/2\right)$$

Answer 2

I think I was able to solve the problem. I remembered

$y = \arcsin x$ is equivalent to $\sin y = x$

which turns the original equation to

$a = (\sin(\arcsin\sqrt b - h/2))^2$