Question

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

Answer 1

Hint: Use the formula $\sin \theta = \cos ({90^ \circ } - \theta )$ and then equate angles on both sides.

Complete step-by-step answer:
From the question,
$ \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 \[67{\dfrac{1}{2}^ \circ }.\]

Note: Whenever we need to solve a trigonometric equation, we try to convert both sides of the equation in the same trigonometric ratio so that we can easily compare their angles.