Question

Find the value of $\theta$ in each of the following

(1) 2 sin 2$\theta$ = $\sqrt{3}$

(2) 2 cos 3$\theta$ = 1

(3) $\sqrt{3}$tan 2$\theta$ - 3 = 0

Answer 1

(1) we have,
2sin 2$\theta$ = $\sqrt{3}$
$\Rightarrow$ sin 2$\theta$ = $\frac{\sqrt{3}}{2}$
$\Rightarrow$ sin 2$\theta$ = sin 2${60}^{\circ }$
$\Rightarrow$ $\theta$ = ${30}^{\circ }$

(2) we have, 2cos 3$\theta$ = 1
$\Rightarrow$ cos3$\theta$ = $\frac{1}{2}$
$\Rightarrow$ coss 3$\theta$ = cos ${60}^{\circ }$
$\Rightarrow$ 3$\theta$ = ${60}^{\circ }$
$\Rightarrow$ $\theta$= ${20}^{\circ }$

(3) we have, $\sqrt{3}$tan 2$\theta$ - 3 = 0 \
$\Rightarrow$ $\sqrt{3}$tan 2$\theta$ = 3
$\Rightarrow$ tan 2$\theta$ = $\frac{3}{\sqrt{3}}$ = $\sqrt{3}$
$\Rightarrow$ tan 2$\theta$ = tan ${60}^{\circ }$ $\Rightarrow$ 2$\theta$ = ${60}^{\circ }$ $\Rightarrow$ $\theta$ = ${30}^{\circ }$