Question

Find general value of $\theta$ which satisfies both $\sin \theta =-1/2$ and $\tan \theta =1/\sqrt{3}$ simultaneously

• A. $\theta =2n\pi +\frac{7\pi }{6}$, $nϵZ$
• B. $\theta =2n\pi +\frac{4\pi }{6}$, $nϵZ$
• C. $\theta =2n\pi +\frac{5\pi }{6}$, $nϵZ$
• D. $\theta =2n\pi +\frac{8\pi }{6}$, $nϵZ$

Answer 1

The correct option is A $\theta =2n\pi +\frac{7\pi }{6}$, $nϵZ$

Here $\sin \theta <0$ and $\tan \theta >0$, then $\theta$ lies in the third quadrant.
Now $\sin \theta =-\frac{1}{2}$ $\Rightarrow$ $\theta =\pi +\frac{\pi }{6}=\frac{7\pi }{6}$
Generalizing, we have $\theta =2n\pi +\frac{7\pi }{6}$, $nϵZ$