Question

$\mathrm{If}\cos 2\theta =\sin 4\theta ,\mathrm{where}2\theta \mathrm{and}4\theta \mathrm{are}\mathrm{acute}\mathrm{angles},\mathrm{then}\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}\theta .$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \cos 2\theta =\sin 4\theta \\ \Rightarrow \sin \left(90°-2\theta \right)=\sin 4\theta \\ \mathrm{Comparing}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ 90°-2\theta =4\theta \\ \Rightarrow 2\theta +4\theta =90° \\ \Rightarrow 6\theta =90° \\ \Rightarrow \theta =\frac{90°}{6} \\ \therefore \theta =15° \end{gathered}$$
Hence, the value of $\theta$ is $15°.$