Question

If \(\sin x\cos x=\frac{1}{4}\), then the general solution of the equation is ...

• A. \(x=\frac{n\pi }{2}+(-1)^{n}\frac{\pi }{6},n\in Z\)
• B. \(x=\frac{n\pi }{2}+(-1)^{n}\frac{\pi }{12},n\in Z\)
• C. \(x=\frac{n\pi }{4}+(-1)^{n}\frac{\pi }{12},n\in Z\)
• D. \(x=\frac{n\pi }{2}+(-1)^{n}\frac{\pi }{3},n\in Z\)

Hint:

Multiply by 2 to get sin 2x = 1/2 and use the general solution of sine.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Use \(2\sin x\cos x=\sin2x\) to convert to a single sine equation.

Step 2: Convert
\(\sin x\cos x=\dfrac14\)
Multiplying by 2: \(\sin2x=\dfrac12=\sin\dfrac{\pi}{6}\).

Step 3: General Solution
\(\sin\theta=\sin\alpha\Rightarrow\theta=n\pi+(-1)^{n}\alpha\)
\(2x=n\pi+(-1)^{n}\dfrac{\pi}{6}\)

Step 4: Solve for x
\(x=\dfrac{n\pi}{2}+(-1)^{n}\dfrac{\pi}{12},\ n\in Z\)
This is option (B).

Final Answer:
The general solution is x equal to n pi over 2 plus minus 1 to the n times pi over 12. \[ \boxed{\text{(B) }x=\dfrac{n\pi}{2}+(-1)^{n}\dfrac{\pi}{12},n\in Z} \]