Question

If $x \in \left(0, \frac{\pi}{2}\right)$ and $x$ satisfies the equation $\sin x \cos x = \frac{1}{4}$, then the values of $x$ are

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

Hint:

Convert the expression to $\sin(2x) = 0.5$ mentally. Knowing that sine hits $0.5$ at $30^\circ$ and $150^\circ$, dividing those angles by 2 gives $15^\circ$ and $75^\circ$ instantly. In radian terms, those are exactly $\frac{\pi}{12}$ and $\frac{5\pi}{12}$!

Answer 1

The Correct Option is A

Step 1: Turn the product into a double angle.
Multiply both sides of $\sin x\cos x=\frac{1}{4}$ by 2. Since $2\sin x\cos x=\sin2x$, we get $\sin2x=\frac{1}{2}$.

Step 2: Find the angles with sine one-half.
$\sin2x=\frac{1}{2}$ gives $2x=\frac{\pi}{6}$ or $2x=\frac{5\pi}{6}$.

Step 3: Solve for $x$.
Dividing by 2, $x=\frac{\pi}{12}$ or $x=\frac{5\pi}{12}$.

Step 4: Check the range.
Both values lie inside $\left(0,\frac{\pi}{2}\right)$, so both are valid. \[ \boxed{x=\frac{\pi}{12},\ \frac{5\pi}{12}} \]