Question

Let $K = \sin \frac{\pi}{18} \sin \frac{5\pi}{18} \sin \frac{7\pi}{18}$ then the value of $\sin \left( 10K \frac{\pi}{3} \right)$ is :

• A. $\frac{\sqrt{3}-1}{2\sqrt{2}}$
• B. $\frac{\sqrt{3}+1}{2\sqrt{2}}$
• C. $\frac{\sqrt{3}+1}{4}$
• D. $\frac{\sqrt{3}-1}{4}$

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We first need to evaluate the value of \( K \) which is a product of sine terms.
Note the angles: \( \frac{\pi}{18} = 10^\circ \), \( \frac{5\pi}{18} = 50^\circ \), \( \frac{7\pi}{18} = 70^\circ \).
Step 2: Key Formula or Approach:
Use the identity: \( \sin \theta \sin(60^\circ - \theta) \sin(60^\circ + \theta) = \frac{1}{4} \sin 3\theta \).
Here \( \theta = 10^\circ \).
Step 3: Detailed Explanation:
\( K = \sin 10^\circ \sin(60^\circ - 10^\circ) \sin(60^\circ + 10^\circ) \)
\( K = \frac{1}{4} \sin(3 \times 10^\circ) = \frac{1}{4} \sin 30^\circ = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} \).
Now find the required value:
\( \sin \left( \frac{10K\pi}{3} \right) = \sin \left( \frac{10 \times \frac{1}{8} \times \pi}{3} \right) = \sin \left( \frac{10\pi}{24} \right) = \sin \left( \frac{5\pi}{12} \right) \).
\( \sin \frac{5\pi}{12} = \sin 75^\circ = \cos 15^\circ \).
\( \cos 15^\circ = \cos(45^\circ - 30^\circ) = \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ \)
\( = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{1}{2} = \frac{\sqrt{3} + 1}{2\sqrt{2}} \).
Step 4: Final Answer:
The value is \( \frac{\sqrt{3}+1}{2\sqrt{2}} \).