Question

Evaluate \[ 4\sin\frac{\pi}{6}\sin\frac{2\pi}{6}\sin\frac{3\pi}{6}\sin\frac{4\pi}{6}\sin\frac{5\pi}{6} \]

• A. \(cos\frac{\pi}{3}cos\frac{2\pi}{3}\)
• B. \(sin\frac{\pi}{3}sin\frac{2\pi}{3}\)
• C. \(sin\frac{\pi}{3}cos\frac{2\pi}{3}\)
• D. \(cos\frac{\pi}{3}sin\frac{2\pi}{3}\)

Hint:

Use identity \(\sin(\pi-\theta)=\sin\theta\) whenever symmetric angles appear.

Answer 1

The Correct Option is B

Concept: Use symmetry of sine function. \[ \sin(\pi-\theta)=\sin\theta \]

Step 1: Simplify terms.
\[ \sin\frac{\pi}{6}=\frac12 \] \[ \sin\frac{2\pi}{6}=\sin\frac{\pi}{3} \] \[ \sin\frac{3\pi}{6}=1 \] \[ \sin\frac{4\pi}{6}=\sin\frac{2\pi}{3} \] \[ \sin\frac{5\pi}{6}=\frac12 \] Expression becomes \[ 4\times\frac12\times\sin\frac{\pi}{3}\times1\times\sin\frac{2\pi}{3}\times\frac12 \]

Step 2: Simplify constants.
\[ 4\times\frac12\times\frac12=1 \] Thus \[ =\sin\frac{\pi}{3}\sin\frac{2\pi}{3} \] Hence \[ \boxed{ \sin\frac{\pi}{3}\sin\frac{2\pi}{3} } \]