Question

For which of the following options, the equality $|\sin x+\cos x|=|\sin x|+|\cos x|$ holds TRUE?

• A. $x$ can be an angle in $\text{I}$ quadrant
• B. $x$ can be an angle in $\text{III}$ quadrant
• C. $x=n\pi ,n\in Z$
• D. $x=(2n+1)\frac{\pi }{2},n\in Z$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $x$ can be an angle in $\text{I}$ quadrant
B $x$ can be an angle in $\text{III}$ quadrant
C $x=n\pi ,n\in Z$
D $x=(2n+1)\frac{\pi }{2},n\in Z$

Since, $|a+b|=|a|+|b|\Rightarrow ab\ge 0$
So, $|\sin x+\cos x|=|\sin x|+|\cos x|$
$\Rightarrow \sin x\cos x\ge 0$

Possible cases are :
$\left(i\right)\sin x$ and $\cos x$ are of same sign.
i.e., $x\in \text{I}$ quadrant or $x\in \text{III}$ quadrant.

$\left(ii\right)\sin x=0\Rightarrow x=n\pi ,n\in Z$

$\left(iii\right)\cos x=0\Rightarrow x=(2n+1)\frac{\pi }{2},n\in Z$