Question

The area bounded by the $x$-axis and the part of graph of $y=\cos x$ between $x=\frac{-\pi}{2}$ and $x=\frac{\pi}{2}$ is separated into two regions by the line $x=k$. If the area of the region for $\frac{-\pi}{2} \leq x \leq k$ is three times the area of the region for $k \leq x \leq \frac{\pi}{2}$, then $k$ is equal to

• A. $\arcsin \left(\frac{1}{4}\right)$
• B. $\arcsin \left(\frac{1}{3}\right)$
• C. $\frac{\pi}{6}$
• D. $\frac{\pi}{3}$

Answer 1

Answer: (C)

$ \int\limits_{-\pi / 2}^{ k } \cos xdx =3 \int\limits_{ k }^{\pi / 2} \cos xdx ; \sin k -\sin \left(-\frac{\pi}{2}\right)=3\left(\sin \frac{\pi}{2}-\sin k \right) $
$\sin k +1=3-3 \sin k ; 4 \sin k =2 \Rightarrow k =\frac{\pi}{6}$