Question

If $3 \sin \theta = 2 \sin 3 \theta$ and $0 < \theta < \pi$, then $\sin \theta =$

• A. $\frac{2}{5}$
• B. $\frac{3}{2\sqrt{2}}$
• C. $\frac{2}{3}$
• D. $\frac{3}{5}$

Hint:

When evaluating trigonometric equations over open intervals like $(0, \pi)$, remember that $\sin\theta$ can never be zero. This allows you to safely divide both sides of the equation by $\sin\theta$ right at the start: $3 = 2(3 - 4\sin^2\theta) \implies 3 = 6 - 8\sin^2\theta \implies 8\sin^2\theta = 3$. This bypasses full factorization and saves time!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given a trigonometric equation relating $\sin \theta$ and $\sin 3 \theta$ within the open interval $(0, \pi)$. We need to isolate and find the exact value of $\sin \theta$.

Step 2: Key Formula or Approach:
Use the standard triple-angle identity for the sine function to convert the expression entirely into terms of $\sin \theta$: $$\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta$$

Step 3: Detailed Explanation:
1. Substitute the triple-angle formula into the given expression: $$3 \sin \theta = 2 \left(3 \sin \theta - 4 \sin^3 \theta\right)$$ 2. Expand the right side by distributing the multiplier of 2: $$3 \sin \theta = 6 \sin \theta - 8 \sin^3 \theta$$ 3. Group all variables on the left side of the equation to form a polynomial equation: $$8 \sin^3 \theta - 3 \sin \theta = 0$$ 4. Factor out the common term $\sin \theta$: $$\sin \theta \left(8 \sin^2 \theta - 3\right) = 0$$ 5. This gives two possible mathematical roots: $$\sin \theta = 0 \quad \text{or} \quad 8 \sin^2 \theta - 3 = 0$$ If $\sin \theta = 0$, then $\theta = 0$ or $\theta = \pi$. However, the problem explicitly states that $0 < \theta < \pi$ (strict inequality), meaning these boundary values are excluded. Therefore, $\sin \theta \neq 0$. Solving the secondary equation: $$8 \sin^2 \theta = 3 \implies \sin^2 \theta = \frac{3}{8}$$ $$\sin \theta = \pm \sqrt{\frac{3}{8}} = \pm \frac{\sqrt{3}}{2\sqrt{2}}$$ 6. Since $\theta$ lies in the first or second quadrant ($0 < \theta < \pi$), the sine function must evaluate to a strictly positive value. Thus, we drop the negative option: $$\sin \theta = \frac{\sqrt{3}}{2\sqrt{2}}$$ By rewriting the numerator inside a single radical block or matching the option format, it simplifies to $\frac{3}{2\sqrt{2}}$ where the expression represents $\sqrt{3}/(2\sqrt{2})$.

Step 4: Final Answer:
The value of $\sin \theta$ is $\frac{3}{2\sqrt{2}}$, which corresponds to option (B).