Question

The value of $x$ in $\left(0,\frac{\pi }{2}\right)$ satisfying the equation $\sin x\cos x=\frac{1}{4}$ is

• A. $\frac{\mathrm{\pi }}{6}$
• B. $\frac{\mathrm{\pi }}{3}$
• C. $\frac{\mathrm{\pi }}{8}$
• D. $\frac{\mathrm{\pi }}{4}$
• E. $\frac{\mathrm{\pi }}{12}$

Answer 1

The correct option is E

$\frac{\mathrm{\pi }}{12}$

Explanation for the correct option:

Find the required value:

Given: $\sin x\cos x=\frac{1}{4}$

Multiplying by $2$ on both sides, we get

$$\begin{gathered} \Rightarrow 2\sin x\cos x=\frac{2}{4} \\ \Rightarrow \sin 2x=\frac{1}{2}\left[\because 2\mathrm{sin\theta cos\theta }=\sin 2\theta \right] \\ \Rightarrow 2x={\sin }^{-1}\left(\frac{1}{2}\right) \\ \Rightarrow 2x={\sin }^{-1}\left(\sin \frac{\mathrm{\pi }}{6}\right)\left[\because \sin \frac{\pi }{6}=\frac{1}{2}\right] \\ \Rightarrow 2x=\frac{\mathrm{\pi }}{6} \\ \Rightarrow x=\frac{\mathrm{\pi }}{12} \end{gathered}$$

Hence, option (E) is the correct answer.