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

Step 1: Read the angles in degrees.
We must evaluate $4\sin\frac{\pi}{6}\sin\frac{2\pi}{6}\sin\frac{3\pi}{6}\sin\frac{4\pi}{6}\sin\frac{5\pi}{6}$. In degrees these angles are $30^\circ, 60^\circ, 90^\circ, 120^\circ, 150^\circ$.
Step 2: Plug in the easy values.
We know $\sin 30^\circ=\frac12$, $\sin 90^\circ=1$, and $\sin 150^\circ=\frac12$. The remaining two are $\sin 60^\circ=\sin\frac{\pi}{3}$ and $\sin 120^\circ=\sin\frac{2\pi}{3}$.
Step 3: Group the constants.
The product becomes $4\cdot\frac12\cdot\sin\frac{\pi}{3}\cdot 1\cdot\sin\frac{2\pi}{3}\cdot\frac12$.
Step 4: Multiply the numeric factors.
Collect $4\cdot\frac12\cdot\frac12 = 1$. So all the leading constants collapse to $1$.
Step 5: Keep the symbolic factors.
What is left is simply $\sin\frac{\pi}{3}\,\sin\frac{2\pi}{3}$, which matches the form of option (2) without needing a numeric value.
Step 6: Sanity check the magnitude.
Numerically $\sin\frac{\pi}{3}=\sin\frac{2\pi}{3}=\frac{\sqrt3}{2}$, so the product equals $\frac34$, which is positive and consistent.
\[ \boxed{\sin\dfrac{\pi}{3}\,\sin\dfrac{2\pi}{3}} \]