Question

The value of \( \sin^{-1}(\sin 10) \) is:

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

Hint:

The graph of $y = \sin^{-1}(\sin x)$ is a "sawtooth" wave that helps identify the correct linear segment for large values of $x$.

Answer 1

The Correct Option is C

Step 1: Concept
$\sin^{-1}(\sin \theta) = \theta$ only if $\theta \in [-\pi/2, \pi/2]$. Otherwise, we must find an angle in the principal range.
Step 2: Analysis
$10$ radians is approximately $10 \times 57.3^{\circ} \approx 573^{\circ}$. We know $3\pi \approx 9.42$. Thus, $10$ is near $3\pi$. $\sin(10) = \sin(10 - 3\pi)$ is not correct because $\sin(3\pi - x) = \sin x$. $\sin(10) = \sin(3\pi - 10)$.
Step 3: Conclusion
$3\pi - 10 \approx 9.42 - 10 = -0.58$, which lies within $[-\pi/2, \pi/2]$. So, $\sin^{-1}(\sin 10) = 3\pi - 10$.
Final Answer: (C)