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: Understanding the Question:
We are given a trigonometric equation $\sin x \cos x = \frac{1}{4}$. We need to determine all valid values of $x$ that satisfy this equation within the open first-quadrant interval $x \in \left(0, \frac{\pi}{2}\right)$.

Step 2: Key Formula or Approach:
We can simplify the left-hand side of the equation by applying the standard sine double-angle identity: $$ 2\sin x \cos x = \sin(2x) $$

Step 3: Detailed Explanation:
Let's multiply both sides of the given equation by 2 to construct the double-angle identity: $$ 2(\sin x \cos x) = 2\left(\frac{1}{4}\right) $$ $$ \sin(2x) = \frac{1}{2} $$ Now, let's look for the principal angles where the sine function equals $\frac{1}{2}$. In the standard domain loops: $$ \sin(2x) = \sin\left(\frac{\pi}{6}\right) \quad \text{or} \quad \sin(2x) = \sin\left(\pi - \frac{\pi}{6}\right) = \sin\left(\frac{5\pi}{6}\right) $$ This sets up two separate cases to isolate $x$:

• Case 1: $$ 2x = \frac{\pi}{6} \implies x = \frac{\pi}{12} $$

• Case 2: $$ 2x = \frac{5\pi}{6} \implies x = \frac{5\pi}{12} $$
Let's double-check if both solutions fall cleanly inside our specified domain interval $\left(0, \frac{\pi}{2}\right)$:

• $\frac{\pi}{12}$ is roughly $15^\circ$, which lies between $0^\circ$ and $90^\circ$.

• $\frac{5\pi}{12}$ is safely less than $\frac{6\pi}{12} = \frac{\pi}{2}$ (roughly $75^\circ$), so it is also valid.
Thus, both values are correct.

Step 4: Final Answer:
The values of $x$ are $\frac{\pi}{12}$ and $\frac{5\pi}{12}$, which corresponds to option (A).