Question:

Step 1: Determine the range and find $x = \sin^{-1}(\sin 10)$. The inverse sine function $\sin^{-1}$ has range $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. We need to find which value in this range has the same sine as $10$ radians. First, note that $3\pi \approx 9.42$ and $4\pi \approx 12.57$, so $10 \in (3\pi, 4\pi)$. We can write: $10 = 3\pi + (10 - 3\pi)$ where $10 - 3\pi \approx 0.575 \in \left(0, \frac{\pi}{2}\right)$. Using the sine addition property: $$\sin(10) = \sin(3\pi + (10-3\pi)) = \sin(3\pi)\cos(10-3\pi) + \cos(3\pi)\sin(10-3\pi)$$ Since $\sin(3\pi) = 0$ and $\cos(3\pi) = -1$: $$\sin(10) = -\sin(10-3\pi)$$ Therefore: $$x = \sin^{-1}(\sin 10) = -(10-3\pi) = 3\pi - 10$$ Step 2: Determine the range and find $y = \cos^{-1}(\cos 10)$. The inverse cosine function $\cos^{-1}$ has range $[0, \pi]$. We need to find which value in this range has the same cosine as $10$ radians. Since $10 \in (3\pi, 4\pi)$, we can write: $10 = 4\pi - (4\pi - 10)$ where $4\pi - 10 \approx 2.57$. Check if $4\pi - 10 \in [0, \pi]$: Since $\pi \approx 3.14$ and $4\pi - 10 \approx 2.57 < \pi$, yes it is in the valid range. Using the cosine property: $$\cos(10) = \cos(4\pi - (4\pi-10)) = \cos(4\pi - 10)$$ Therefore: $$y = \cos^{-1}(\cos 10) = 4\pi - 10$$ Step 3: Calculate $y - x$. Now we compute the difference: $$y - x = (4\pi - 10) - (3\pi - 10) = 4\pi - 10 - 3\pi + 10 = \pi$$ **Final Answer:** $y - x = \pi$ The answer is **Option 1: $\pi$** <div class="key-concept"><strong>Key Concept:</strong> Principal value branches of inverse trigonometric functions.</div> <div class="trap-box"><strong>Trap:</strong> Students may forget to properly reduce the angle to the principal value range for both functions, leading to sign errors.</div>