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: Simplify $K$.
The angles are in radians. Convert to degrees for convenience:
$\frac{\pi}{18} = 10^\circ, \frac{5\pi}{18} = 50^\circ, \frac{7\pi}{18} = 70^\circ$.
$K = \sin 10^\circ \sin 50^\circ \sin 70^\circ$.
Using the identity $\sin \theta \sin(60-\theta) \sin(60+\theta) = \frac{1}{4} \sin 3\theta$:
For $\theta = 10^\circ$, we have $60-\theta = 50^\circ$ and $60+\theta = 70^\circ$.
So, $K = \frac{1}{4} \sin(3 \times 10^\circ) = \frac{1}{4} \sin 30^\circ = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$.


Step 2: Calculate the required expression.
We need $\sin(10 K \frac{\pi}{3})$:
Substitute $K = 1/8$:
$\sin \left( 10 \cdot \frac{1}{8} \cdot \frac{\pi}{3} \right) = \sin \left( \frac{10\pi}{24} \right) = \sin \left( \frac{5\pi}{12} \right)$

Step 3: Evaluate $\sin \frac{5\pi}{12}$.
$\frac{5\pi}{12} = 75^\circ$.
$\sin 75^\circ = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ$
$= \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{1}{2} = \frac{\sqrt{3} + 1}{2\sqrt{2}}$
This matches option (2).