Question

If $2\theta +{45}^{\circ }and{30}^{\circ }-\theta$ are acute angles, find the degree measure of $\theta$ satisfying $\sin (2\theta +{45}^{\circ })=\cos ({30}^{\circ }-\theta )$

Answer 1

Given,

$2\theta +{45}^{\circ }$ and $({30}^{\circ }-\theta )$ are acute and $\sin (2\theta +{45}^{\circ })=\cos ({30}^{\circ }-\theta )$

We have, $\cos \theta =\sin ({90}^{\circ }-\theta )$

$\cos ({30}^{\circ }-\theta )=\sin ({90}^{\circ }-({30}^{\circ }-\theta \left)\right)=\sin ({60}^{\circ }+\theta )$

$\therefore \sin (2\theta +{45}^{\circ })=\sin ({60}^{\circ }+\theta )$

$2\theta +{45}^{\circ }={60}^{\circ }+\theta$

$2\theta -\theta ={60}^{\circ }-{45}^{\circ }$

$\theta ={15}^{\circ }$