Question

The value(s) of $x$ satisfying both the equations $\sin x=-\frac{1}{2}$ and $\tan x=\frac{1}{\sqrt{3}}$ in $[0,2\pi ]$ simultaneously is :

• A. $\frac{\pi }{6}$
• B. $\frac{7\pi }{6}$
• C. $\frac{11\pi }{6}$
• D. $\frac{7\pi }{6},\frac{11\pi }{6}$

Answer 1

The correct option is B $\frac{7\pi }{6}$

$\because \sin x<0$ and $\tan x>0$
$\therefore x$ is in third quadrant
$\sin x=-\frac{1}{2}\Rightarrow x=\frac{7\pi }{6},\frac{11\pi }{6}$
and
$\tan x=\frac{1}{\sqrt{3}}\Rightarrow x=\frac{\pi }{6},\frac{7\pi }{6}$
$\therefore$ Common solution $=\frac{7\pi }{6}$