Question

The value of ${\sin }^{-1}\left[\sin \left(10\right)\right]$ is

• A. $10$
• B. $10-3\mathrm{\pi }$
• C. $3\mathrm{\pi }-10$
• D. None of these.

Answer 1

The correct option is C

$3\mathrm{\pi }-10$

Explanation for the correct option:

The given trigonometric expression: ${\sin }^{-1}\left[\sin \left(10\right)\right]$.

It is known that, ${\sin }^{-1}\left(\theta \right)\in \left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]$.Thus, $10\mathrm{rad}\in \left(3\mathrm{\pi },\frac{7\mathrm{\pi }}{2}\right)$.

Let us assume that, $10\mathrm{rad}=3\mathrm{\pi }+\mathrm{\varphi }$.

$\Rightarrow \mathrm{\varphi }=10-3\mathrm{\pi }$.

Thus, ${\sin }^{-1}\left[\sin \left(10\right)\right]={\sin }^{-1}\left[\sin \left(3\mathrm{\pi }+\mathrm{\varphi }\right)\right]$

$$\begin{gathered} \Rightarrow {\sin }^{-1}\left[\sin \left(10\right)\right]={\sin }^{-1}\left[-\sin \left(\mathrm{\varphi }\right)\right]\left[\because \sin \left(3\mathrm{\pi }+\mathrm{\theta }\right)=-\sin \left(\theta \right)\right] \\ \Rightarrow {\sin }^{-1}\left[\sin \left(10\right)\right]={\sin }^{-1}\left[\sin \left(-\mathrm{\varphi }\right)\right] \\ \Rightarrow {\sin }^{-1}\left[\sin \left(10\right)\right]={\sin }^{-1}\left[\sin \left(3\mathrm{\pi }-10\right)\right]\left[\because \mathrm{\varphi }=10-3\pi \right] \\ \Rightarrow {\sin }^{-1}\left[\sin \left(10\right)\right]=\left(3\mathrm{\pi }-10\right) \end{gathered}$$

Therefore, the value of ${\sin }^{-1}\left[\sin \left(10\right)\right]$ is $3\mathrm{\pi }-10$.

Hence, the correct option is (C).