Question

If $\sin 5\theta =\cos 4\theta$, where $5\theta$ and $4\theta$ are acute angles, find the value of $\theta$.

Answer 1

Consider $\sin 5\theta$ which can be rewritten as follows:

$\sin 5\theta =\cos ({90}^0-5\theta )$

$(\because \cos ({90}^0-x)=\sin x)$

Therefore, $\sin 5\theta =\cos 4\theta$ can be evaluated as shown below:

$\sin 5\theta =\cos 4\theta \Rightarrow \cos ({90}^0-5\theta )=\cos 4\theta \Rightarrow {90}^0-5\theta =4\theta \Rightarrow {90}^0=4\theta +5\theta \Rightarrow {90}^0=9\theta \Rightarrow \theta =\frac{{90}^0}{9}={10}^0$

Hence, $\theta ={10}^0$.