Question

2sin(\(\frac{\pi}{22}\))sin(\(\frac{3\pi}{22}\))sin(\(\frac{5\pi}{22}\))sin(\(\frac{7\pi}{22}\))sin(\(\frac{9\pi}{22}\)) is equal to

• A. \(\frac{3}{16}\)
• B. \(\frac{1}{16}\)
• C. \(\frac{1}{32}\)
• D. \(\frac{9}{32}\)

Answer 1

The Correct Option is B

To solve the problem, we need to evaluate the expression:

\(2 \cdot \sin\left(\frac{\pi}{22}\right)\sin\left(\frac{3\pi}{22}\right)\sin\left(\frac{5\pi}{22}\right)\sin\left(\frac{7\pi}{22}\right)\sin\left(\frac{9\pi}{22}\right)\). 

This expression is a classic problem involving the product of sine terms that can be solved using trigonometric identities and symmetry properties of the sine function.

We can use a trigonometric identity known as the multiple angle identity for sine, which states:

\(\prod_{k=1}^{n} \sin\left(\frac{k\pi}{2n+1}\right) = \frac{1}{2^n}\), for \(n = 5\).

Applying this identity to our problem, the product of sine terms:

\(\sin\left(\frac{\pi}{22}\right)\sin\left(\frac{3\pi}{22}\right)\sin\left(\frac{5\pi}{22}\right)\sin\left(\frac{7\pi}{22}\right)\sin\left(\frac{9\pi}{22}\right)\)

results in:

\(\frac{1}{2^5}\) which simplifies to \(\frac{1}{32}\).

Since the expression is multiplied by 2, we have:

\(2 \times \frac{1}{32} = \frac{1}{16}\).

Hence, the evaluated expression is \(\frac{1}{16}\).

Therefore, the correct answer is \(\frac{1}{16}\), which matches option B.

Answer 2

\[ 2 \sin \left( \frac{\pi}{22} \right) \sin \left( \frac{3\pi}{22} \right) \sin \left( \frac{5\pi}{22} \right) \sin \left( \frac{7\pi}{22} \right) \sin \left( \frac{9\pi}{22} \right) \]  
Using the symmetry of angles: \[ = 2 \sin \left( \frac{11\pi - 10\pi}{22} \right) \sin \left( \frac{11\pi - 8\pi}{22} \right) \sin \left( \frac{11\pi - 6\pi}{22} \right) \sin \left( \frac{11\pi - 4\pi}{22} \right) \sin \left( \frac{11\pi - 2\pi}{22} \right) \] 
Simplifying: \[ = 2 \cdot \frac{\cos \pi}{11} \cdot \frac{2\cos \pi}{11} \cdot \frac{\cos 3\pi}{11} \cdot \frac{\cos 4\pi}{11} \cdot \frac{\cos 5\pi}{11} \] 
\[ = \frac{2 \sin \left( \frac{32\pi}{11} \right)}{2^5 \sin \left( \frac{\pi}{11} \right)} \] 
Finally: \[ = \frac{1}{16} \]