If $y=\sin^2(a+\delta)$, then is there an expression for $\sin^2\delta$ in terms of $a$ and $y$?

Question

Consider this equation: $$y = \sin^2(a+\delta)$$

Can I get the expression of $\sin^2 \delta$ from this by any trigonometric manipulations? or is this a transcendental equation and can only be solved numerically?

Answer 1

Look at the picture.

$$ \begin{align} |OA|&=1\\ |AB|&=\sin{(a+\delta)}\\ &=\sqrt{y}\\ |OB|&=\cos{(a+\delta)}\\ &=\sqrt{1-y}\\ \\ |AP|&=|AB|-|BP|\\ &=\sqrt{y}-\sqrt{1-y}\tan{(a)}\\ \\ |AQ|&=|AP|\sin{(90-a)}\\ &=\left(\sqrt{y}-\sqrt{1-y}\tan{(a)}\right)\cdot \cos{(a)} \end{align} $$

Notice that $|AQ|=\sin{(\delta)}$. Try for different quadrants as well.

Answer 2

$$y = \sin^2(a+\delta)=\frac{1-\cos[2(a+\delta)]}2\implies \cos[2(a+\delta)]=1-2y$$ Using the inverse trigonomatric functions $$\delta=-a\pm\frac{1}{2} \cos ^{-1}(1-2 y)$$ $$\sin^2(\delta)=\sin^2\left(a\pm\frac{1}{2} \cos ^{-1}(1-2 y) \right)$$