Question

\left\\{x\,\in\,R : \left| \cos\,x\right|\ge \sin\,x\right\\} \cap\left[0, \frac{3\pi}{2}\right]=

• A. $\left[0, \frac{\pi}{4}\right]\cup\left[\frac{3\pi}{4}, \frac{3\pi}{2}\right]$
• B. $\left[0, \frac{\pi}{4}\right]\cup\left[\frac{\pi}{2}, \frac{3\pi}{2}\right]$
• C. $\left[0, \frac{\pi}{4}\right]\cup\left[\frac{5\pi}{4}, \frac{3\pi}{2}\right]$
• D. $\left[0, \frac{3\pi}{2}\right]$

Answer 1

Answer: (A)

$\left[0, \frac{\pi}{4}\right]\cup\left[\frac{3\pi}{4}, \frac{3\pi}{2}\right]$

Answer 2

Given, $\\{x \in R:|\cos x| \geq \sin x\\} \cap\left[0, \frac{3 \pi}{2}\right]$
If we draw the graphs of $|\cos x|$ and $\sin x$, clearly
$|\cos x| \geq \sin x$ when

$x \in\left[0, \frac{\pi}{4}\right] \cup\left[\frac{3 \pi}{4}, \frac{3 \pi}{2}\right]$
$\therefore x \in\left[0, \frac{\pi}{4}\right] \cup\left[\frac{3 \pi}{4}, \frac{3 \pi}{2}\right] \cap\left[0, \frac{3 \pi}{2}\right]$
$\Rightarrow x \in\left[0, \frac{\pi}{4}\right] \cup\left[\frac{3 \pi}{4}, \frac{3 \pi}{2}\right]$