Question

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

Answer 1

$\sin \theta =-\frac{1}{2}$ and $\tan \theta =\frac{1}{\sqrt{3}}$

$\Rightarrow \theta =\{-\frac{\pi }{6},2\pi -\frac{\pi }{6},\pi +\frac{\pi }{6}\}\cap \{\frac{\pi }{6},\pi +\frac{\pi }{6}\}$

$\Rightarrow \theta =\{-\frac{\pi }{6},\frac{11\pi }{6},\frac{7\pi }{6}\}\cap \{\frac{\pi }{6},\frac{7\pi }{6}\}$

$\Rightarrow \theta =2n\pi +\frac{7\pi }{6}$