Question

Solve:
${\sin }^{-1}\left(\frac{a\sin x+b\cos x}{\sqrt{a^2+b^2}}\right)$

Answer 1

$\cos \theta =\frac{a}{\sqrt{a^2+b^2}}$

$\sin \theta =\frac{b}{\sqrt{a^2+b^2}}$

$={\sin }^{-1}(\cos \theta \sin x+\sin \theta \cos x)$

$={\sin }^{-1}\left(\sin \right(\theta +x\left)\right)$

$=\theta +x$

$=x+{\sin }^{-1}\frac{b}{\sqrt{a^2+b^2}}$