Question

The value of \(2 \sin(\frac{\pi}{8}) \sin(\frac{2\pi}{8}) \sin(\frac{3\pi}{8}) \sin(\frac{5\pi}{8}) \sin(\frac{6\pi}{8}) \sin(\frac{7\pi}{8})\) is :

• A. \(\frac{1}{8}\)
• B. \(\frac{1}{8\sqrt{2}}\)
• C. \(\frac{1}{4}\)
• D. \(\frac{1}{4\sqrt{2}}\)

Hint:

For products of sine or cosine functions with arguments in an arithmetic progression, always look for symmetries using identities like \(\sin(\pi-\theta)\), \(\cos(\pi-\theta)\), \(\sin(\pi/2-\theta)\), etc. This often allows you to pair up terms and simplify the expression significantly.

Answer 1

The Correct Option is A

Step 1: Understanding the Question
We need to evaluate a product of sine functions with arguments that are multiples of \(\pi/8\).
Step 2: Key Formula or Approach
We will use trigonometric identities to simplify the product. Key identities: \(\sin(\pi - \theta) = \sin\theta\) \(\sin(\frac{\pi}{2} - \theta) = \cos\theta\) \(\sin(2\theta) = 2\sin\theta\cos\theta\)
Step 3: Detailed Explanation
Let the expression be E. \[ E = 2 \sin(\frac{\pi}{8}) \sin(\frac{2\pi}{8}) \sin(\frac{3\pi}{8}) \sin(\frac{5\pi}{8}) \sin(\frac{6\pi}{8}) \sin(\frac{7\pi}{8}) \] First, use the identity \(\sin(\pi - \theta) = \sin\theta\): \(\sin(\frac{7\pi}{8}) = \sin(\pi - \frac{\pi}{8}) = \sin(\frac{\pi}{8})\) \(\sin(\frac{6\pi}{8}) = \sin(\pi - \frac{2\pi}{8}) = \sin(\frac{2\pi}{8})\) \(\sin(\frac{5\pi}{8}) = \sin(\pi - \frac{3\pi}{8}) = \sin(\frac{3\pi}{8})\) Substituting these back into the expression: \[ E = 2 [\sin(\frac{\pi}{8}) \sin(\frac{2\pi}{8}) \sin(\frac{3\pi}{8})]^2 \] Now, use the identity \(\sin(\frac{\pi}{2} - \theta) = \cos\theta\). Note that \(\frac{\pi}{2} = \frac{4\pi}{8}\). \(\sin(\frac{3\pi}{8}) = \sin(\frac{4\pi}{8} - \frac{\pi}{8}) = \sin(\frac{\pi}{2} - \frac{\pi}{8}) = \cos(\frac{\pi}{8})\) Also, \(\sin(\frac{2\pi}{8}) = \sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}\). Substituting these gives: \[ E = 2 \left[\sin(\frac{\pi}{8}) \cdot \frac{1}{\sqrt{2}} \cdot \cos(\frac{\pi}{8})\right]^2 \] \[ E = 2 \left[ \frac{1}{\sqrt{2}} \left(\sin(\frac{\pi}{8})\cos(\frac{\pi}{8})\right) \right]^2 \] Now, use the identity \(2\sin\theta\cos\theta = \sin(2\theta)\), which means \(\sin\theta\cos\theta = \frac{1}{2}\sin(2\theta)\). \[ E = 2 \left[ \frac{1}{\sqrt{2}} \cdot \frac{1}{2}\sin\left(2 \cdot \frac{\pi}{8}\right) \right]^2 = 2 \left[ \frac{1}{2\sqrt{2}}\sin\left(\frac{\pi}{4}\right) \right]^2 \] Substitute \(\sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}\): \[ E = 2 \left[ \frac{1}{2\sqrt{2}} \cdot \frac{1}{\sqrt{2}} \right]^2 = 2 \left[ \frac{1}{4} \right]^2 = 2 \cdot \frac{1}{16} = \frac{1}{8} \] Step 4: Final Answer
The value of the expression is \(\frac{1}{8}\).