Question

Which of the following is not the solution of the equation $\sin 5x = 16 \sin^{5} x (n \in Z)?$

• A. $n\pi + \frac{\pi}{6}$
• B. $n\pi - \frac{\pi}{6}$
• C. $n\pi$
• D. $n\pi + \frac{\pi}{3}$

Hint:

Check the given options by substituting basic values (like $n=0$) into the original equation to see if they satisfy it.

Answer 1

The Correct Option is D

Step 1: Concept
Use the trigonometric identity for $\sin 5x$ and solve the resulting equation for $x$.

Step 2: Meaning
$\sin 5x = 5 \sin x - 20 \sin^{3} x + 16 \sin^{5} x$.

Step 3: Analysis
Setting $\sin 5x = 16 \sin^{5} x$ gives $5 \sin x - 20 \sin^{3} x = 0 \implies 5 \sin x (1 - 4 \sin^{2} x) = 0$. This implies $\sin x = 0$ or $\sin x = \pm 1/2$.

Step 4: Conclusion
$\sin x = 0 \implies x = n\pi$. $\sin x = \pm 1/2 \implies x = n\pi \pm \pi/6$. The value $n\pi + \pi/3$ is not a solution. Final Answer: (D)