Question

The values of x in $(0,\frac{\pi }{2})$ satisfying the equation $\sin x\cos x=\frac{1}{4}$ are ________.

• A. $\frac{\pi }{6},\frac{\pi }{12}$
• B. $\frac{\pi }{12},\frac{5\pi }{12}$
• C. $\frac{\pi }{8},\frac{3\pi }{8}$
• D. $\frac{\pi }{8},\frac{\pi }{4}$

Answer 1

The correct option is B $\frac{\pi }{12},\frac{5\pi }{12}$

We know that, $\text{sin}\left(2x\right)=2\text{sin}x\text{cos}x$.

So, from $\text{sin}x\text{cos}x=\frac{1}{4}$, we get that $\text{sin}\left(2x\right)=\frac{1}{2}$.

So, we have to find the solutions of the equation $\text{sin}\left(2x\right)=\frac{1}{2}$, where $x\in (0,\frac{\pi }{2})$.

From this data, we get that $2x=\frac{\pi }{6}$ or $\frac{5\pi }{6}$.

That is, $x=\frac{\pi }{12}$ or $\frac{5\pi }{12}$.