Question

If $\sin x \cos x = \frac{1}{4}$, then the general solution is:

• A. $x = n\pi + (-1)^n \frac{\pi}{6}$
• B. $x = \frac{n\pi}{2} + (-1)^n \frac{\pi}{12}$
• C. $x = \frac{n\pi}{2} + \frac{\pi}{12}$
• D. $x = n\pi \pm \frac{\pi}{6}$

Hint:

Multiplying by 2 to create $\sin 2x$ is the standard first step for trigonometric products of sine and cosine.

Answer 1

The Correct Option is B

Step 1: Concept
Use the double angle formula for sine, $\sin 2x = 2 \sin x \cos x$, to simplify the product of trigonometric terms.

Step 2: Meaning
The given equation $\sin x \cos x = 1/4$ can be multiplied by 2 on both sides to give $2 \sin x \cos x = 1/2$, which simplifies to $\sin 2x = 1/2$.

Step 3: Analysis
The general solution for $\sin \theta = \sin \alpha$ is $\theta = n\pi + (-1)^n \alpha$. Since $\sin(\pi/6) = 1/2$, we substitute $\theta = 2x$ and $\alpha = \pi/6$ to get $2x = n\pi + (-1)^n (\pi/6)$.

Step 4: Conclusion
Dividing the entire equation by 2 yields the general solution $x = \frac{n\pi}{2} + (-1)^n \frac{\pi}{12}$. Final Answer: (B)