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Question: The fourth question presented is: \(\frac{\sin \frac{\pi }{14}}{\sin \frac{3\pi }{14}}\frac{\sin \fr...

The fourth question presented is: sin⁡π14sin⁡3π14sin⁡5π14sin⁡7π14sin⁡9π14sin⁡11π14sin⁡13π14sin⁡14π14\frac{\sin \frac{\pi }{14}}{\sin \frac{3\pi }{14}}\frac{\sin \frac{5\pi }{14}}{\sin \frac{7\pi }{14}}\frac{\sin \frac{9\pi }{14}}{\sin \frac{11\pi }{14}}\frac{\sin \frac{13\pi }{14}}{\sin \frac{14\pi }{14}}

Answer

Undefined

Explanation

Solution

The given expression is: E=sin⁡π14sin⁡3π14sin⁡5π14sin⁡7π14sin⁡9π14sin⁡11π14sin⁡13π14sin⁡14π14E = \frac{\sin \frac{\pi }{14}}{\sin \frac{3\pi }{14}}\frac{\sin \frac{5\pi }{14}}{\sin \frac{7\pi }{14}}\frac{\sin \frac{9\pi }{14}}{\sin \frac{11\pi }{14}}\frac{\sin \frac{13\pi }{14}}{\sin \frac{14\pi }{14}}

Let's analyze the terms in the expression. We know that sin⁡(π−x)=sin⁡x\sin(\pi - x) = \sin x. Using this identity:

sin⁡13π14=sin⁡(π−π14)=sin⁡π14\sin \frac{13\pi}{14} = \sin \left(\pi - \frac{\pi}{14}\right) = \sin \frac{\pi}{14}

sin⁡11π14=sin⁡(π−3π14)=sin⁡3π14\sin \frac{11\pi}{14} = \sin \left(\pi - \frac{3\pi}{14}\right) = \sin \frac{3\pi}{14}

sin⁡9π14=sin⁡(π−5π14)=sin⁡5π14\sin \frac{9\pi}{14} = \sin \left(\pi - \frac{5\pi}{14}\right) = \sin \frac{5\pi}{14}

Also, sin⁡7π14=sin⁡π2=1\sin \frac{7\pi}{14} = \sin \frac{\pi}{2} = 1. And sin⁡14π14=sin⁡π=0\sin \frac{14\pi}{14} = \sin \pi = 0.

Substitute these values into the expression: E=sin⁡π14sin⁡3π14sin⁡5π141sin⁡5π14sin⁡3π14sin⁡π140E = \frac{\sin \frac{\pi }{14}}{\sin \frac{3\pi }{14}}\frac{\sin \frac{5\pi }{14}}{1}\frac{\sin \frac{5\pi }{14}}{\sin \frac{3\pi }{14}}\frac{\sin \frac{\pi }{14}}{0}

Since the last term has sin⁡π=0\sin \pi = 0 in the denominator, the entire expression is undefined.

In mathematics, division by zero is undefined. Therefore, the value of the given expression is undefined.