Question

If $3\theta$ is an acute angle, solve the following equations for $\theta$:
(i)$2\sin 3\theta =\sqrt{3}$
(ii)$\tan 3\theta =1.$

Answer 1

$\left(i\right)2\sin 3\theta =\sqrt{3}$

$\Rightarrow \sin 3\theta =\frac{\sqrt{3}}{2}$

$\Rightarrow 3\theta =\frac{\pi }{3},\pi -\frac{\pi }{3}$

$\Rightarrow 3\theta =\frac{\pi }{3},\frac{2\pi }{3}$

$\therefore \theta =\frac{\pi }{9},\frac{2\pi }{9}$

$\left(ii\right)\tan 3\theta =1$

$\Rightarrow 3\theta =\frac{\pi }{4},\pi +\frac{\pi }{4}$

$\therefore \theta =\frac{\pi }{12},\frac{5\pi }{12}$