Question

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

Answer 1

Given that $\sin \theta =\frac{-1}{2}$

We know that $\sin \theta$ is negative in $Q_3$ and $Q_4$

Given that $\tan \theta =\frac{1}{\sqrt{3}}$

We know that $\tan \theta$ is positive in $Q_1$ and $Q_3$

Thus we can say that $\pi <\theta <\frac{3\pi }{2}$

From given values of $\sin \theta$ and $\tan \theta$ we can say that $\theta =\frac{5\pi }{4}$

Therefore, general solution $\Rightarrow \theta =(2n+1)\pi +\frac{\pi }{4}$