Question

If $0\le x\le 2\pi$ and $|\cos x|\le \sin x$, then

• A. The set values of x is $[\frac{\pi }{4},\frac{3\pi }{4}]$
• B. then number of solutions thta are integral multiple of $\frac{\pi }{4}$ is four.
• C. the sum of the largest and the smallest solution is $\pi$
• D. the set of all values of x is $xϵ[\frac{\pi }{4},\frac{\pi }{2})\cup (\frac{\pi }{2},3\pi 4]$

Answer 1

Answer: (A), (C)

The correct options are
A The set values of x is $[\frac{\pi }{4},\frac{3\pi }{4}]$
C the sum of the largest and the smallest solution is $\pi$

$\mid \cos x\mid \le \sin x$, $0\le x\le 2\pi$
If $0\le x\le \frac{\pi }{2}\Rightarrow \cos x\ge 0\Rightarrow \mid \cos x\mid =\cos x$
$\Rightarrow \tan x\ge 1\Rightarrow xϵ[\frac{\pi }{4},\frac{\pi }{2}]$
If $\frac{\pi }{2}<x\le \pi \Rightarrow \cos x\le 0\Rightarrow \mid \cos x\mid =-\cos x$
$\Rightarrow \tan x\le 1\Rightarrow xϵ(\frac{\pi }{2},\frac{3\pi }{4}]$
considering both cases,
$xϵ[\frac{\pi }{4},\frac{\pi }{2}]\cup (\frac{\pi }{2},\frac{3\pi }{4}]$
Hence option 'C' is correct.
second method $\Rightarrow$ It is easy if you solve it by graphical method.