QUESTION

Find the degree measure of angle $\theta$, if $\sin \theta = \cos (\theta - 45),$ where $\theta$ and $\theta - {45^ \circ }$ are acute angles.

Hint: Use the formula $\sin \theta = \cos ({90^ \circ } - \theta )$ and then equate angles on both sides.
$\Rightarrow \sin \theta = \cos (\theta - 45){\text{ }}.....(i)$
We know that, $\sin \theta = \cos (90 - \theta ),$putting this value in equation $(i)$:
$\Rightarrow \cos ({90^ \circ } - \theta ) = \cos (\theta - {45^ \circ }), \\ \Rightarrow {90^ \circ } - \theta = \theta - {45^ \circ }, \\ \Rightarrow 2\theta = {135^ \circ }, \\ \Rightarrow \theta = \dfrac{{{{135}^ \circ }}}{2} = 67{\dfrac{1}{2}^ \circ } \\$
Therefore, the value of angle $\theta$ is $67{\dfrac{1}{2}^ \circ }.$