Question

If sin \( \theta \) = \( \frac{1}{2}\left(x + \frac{1}{x}\right) \), then sin \( 3\theta + \frac{1}{2}\left(x^3 + \frac{1}{x^3}\right) \) =}

• A. 0
• B. 1
• C. \( \frac{1}{4} \)
• D. 2

Hint:

$x + 1/x$ is always $\ge 2$ or $\le -2$. This often forces $x$ to be 1 or $-1$ in trig equations.

Answer 1

The Correct Option is A

Step 1: Identify constraint
For any real $x$, $|x + \frac{1}{x}| \ge 2$. Thus $\frac{1}{2}|x + \frac{1}{x}| \ge 1$.
Since $|\sin \theta| \le 1$, the only possible value is $\sin \theta = \pm 1$, which occurs when $x = \pm 1$.
Step 2: Case Analysis
If $x = 1$, $\sin \theta = 1$. Then $\theta = \frac{\pi}{2}$.
$\sin 3\theta = \sin(\frac{3\pi}{2}) = -1$.
Step 3: Calculate Expression
$-1 + \frac{1}{2}(1^3 + \frac{1}{1^3}) = -1 + \frac{1}{2}(2) = -1 + 1 = 0$.
Step 4: Conclusion
The value is 0.
Final Answer:(A)