Question:
If $\sin \theta=\cos \left(\theta-45^{\circ}\right)$, where $\theta$ and $\theta-45^{\circ}$ are acute angles, find the degree measure of $\theta .$
Solution:
Given that: $\sin \theta=\cos \left(\theta-45^{\circ}\right)$ where $\theta$ and $\left(\theta-45^{\circ}\right)$ are acute angles
We have to find $\theta$
$\sin \theta=\cos \left(\theta-45^{\circ}\right)$
$\Rightarrow \cos \left(90^{\circ}-\theta\right)=\cos \left(\theta-45^{\circ}\right)$
$\Rightarrow 90^{\circ}-\theta=\theta-45^{\circ}$
$\Rightarrow-2 \theta=-125^{\circ}$
$\Rightarrow \theta=\frac{135^{\circ}}{2}$
Therefore $\theta=67 \frac{1}{2}$