Question

Find $\theta$ if
1. $\tan 3\theta =\sin {45}^{\circ }\cos {45}^{\circ }+\sin {30}^{\circ }$
2. $\sin 2\theta =\sin {60}^{\circ }\cos {30}^{\circ }-\cos {60}^{\circ }\sin {30}^{\circ }$

Answer 1

$\left(1\right)$

$\begin{array}{c}\tan 3\theta =\sin {45}^{\circ }.\cos {45}^{\circ }+\sin {30}^{\circ }\\ =\frac{1}{\sqrt{2}}.\frac{1}{\sqrt{2}}+\frac{1}{2}\\ =\frac{1}{2}+\frac{1}{2}=1\\ \tan 3\theta =1\\ 3\theta ={45}^{\circ },\\ \theta ={15}^{\circ }\end{array}$

$\left(2\right)$

$\begin{array}{c}\sin 2\theta =\sin {60}^{\circ }.\cos {30}^{\circ }-cos{60}^{\circ }.\sin {30}^{\circ }\\ =\sin ({60}^{\circ }-{30}^{\circ })\\ =\sin {30}^{\circ }\\ 2\theta ={30}^{\circ }\\ \theta ={15}^{\circ }\end{array}$