Question

The principal value of $\sin^{-1} \left( \sin \left( -\frac{10\pi}{3} \right) \right)$ is :

• A. $-\frac{2\pi}{3}$
• B. $-\frac{\pi}{3}$
• C. $\frac{\pi}{3}$
• D. $\frac{2\pi}{3}$

Hint:

For sine inverse problems, always simplify the argument first to bring it within the range of the principal value, which is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

Answer 1

The Correct Option is B

To address this, we begin by simplifying the argument of the sine function: \[ \sin \left( -\frac{10\pi}{3} \right) = \sin \left( -\frac{10\pi}{3} + 2\pi \times 2 \right) = \sin \left( -\frac{10\pi}{3} + \frac{12\pi}{3} \right) = \sin \left( \frac{2\pi}{3} \right) \] Subsequently, we evaluate $\sin^{-1} \left( \sin \left( \frac{2\pi}{3} \right) \right)$. This results in the principal value of $-\frac{\pi}{3}$, because the output of the arcsine function is restricted to the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$.