Trigonometry & Inverse Trigonometry
Inverse Trig Functions
nta_abhyas_2025
Grade None
Question:
The area bounded by the curve $y = |\cos^{-1}(\sin x)| + |\frac{\pi}{2} - \cos^{-1}(\cos x)|$ and the $x$-axis, where $\frac{\pi}{2} \leq x \leq \pi$, is equal to
\frac{\pi^2}{8}
\frac{\pi^2}{4}
\frac{\pi^2}{2}
\frac{7\pi^2}{16}
Step-by-Step Solution
Key Concept: Evaluate the absolute values and inverse trig functions carefully over the given domain to find the constant function
For $x \in [\frac{\pi}{2}, \pi]$, we have $\sin x \geq 0$ so $|\sin x| = \sin x$, and $\cos x \leq 0$ so $|\cos x| = -\cos x$. Thus $y = |\frac{\pi}{2} - \sin^{-1}(\sin x)| + |\cos^{-1}(-\cos x)|$. This simplifies to $y = |\frac{\pi}{2} - (\pi - x)| + |\pi - x| = |x - \frac{\pi}{2}| + (\pi - x) = (x - \frac{\pi}{2}) + (\pi - x) = \frac{\pi}{2}$. The area is $\frac{1}{2} \times \frac{\pi}{2} \times \frac{\pi}{2} = \frac{\pi^2}{4}$.
Correct Answer: π²/4