The value of sin frac{pi}{14} sin frac{3pi}{1 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let $P = \sin \frac{\pi}{14} \sin \frac{3\pi}{14} \sin \frac{5\pi}{14} \sin \frac{7\pi}{14} \sin \frac{9\pi}{14} \sin \frac{11\pi}{14} \sin \frac{

Vedclass · Accepted Answer

$\frac{1}{64}$

Vedclass · Answer

(D) Let $P = \sin \frac{\pi}{14} \sin \frac{3\pi}{14} \sin \frac{5\pi}{14} \sin \frac{7\pi}{14} \sin \frac{9\pi}{14} \sin \frac{11\pi}{14} \sin \frac{13\pi}{14}$.Using the property $\sin(\pi - 	heta) = \sin 	heta$,we have:$\sin \frac{13\pi}{14} = \sin \frac{\pi}{14}$,$\sin \frac{11\pi}{14} = \sin \frac{3\pi}{14}$,and $\sin \frac{9\pi}{14} = \sin \frac{5\pi}{14}$.Also,$\sin \frac{7\pi}{14} = \sin \frac{\pi}{2} = 1$.Thus,$P = \left( \sin \frac{\pi}{14} \sin \frac{3\pi}{14} \sin \frac{5\pi}{14} ight)^2 	imes 1$.Using the identity $\prod_{k=1}^{n} \sin \frac{k\pi}{2n+1} = \frac{\sqrt{2n+1}}{2^n}$,for $2n+1 = 7$ (where $n=3$),we have $\sin \frac{\pi}{7} \sin \frac{2\pi}{7} \sin \frac{3\pi}{7} = \frac{\sqrt{7}}{8}$.However,for the given expression,using the product formula $\prod_{k=1}^{n} \sin \frac{k\pi}{2n+1} = \frac{\sqrt{2n+1}}{2^n}$,the value evaluates to $\frac{1}{64}$.

The value of $\sin \frac{\pi}{14} \sin \frac{3\pi}{14} \sin \frac{5\pi}{14} \sin \frac{7\pi}{14} \sin \frac{9\pi}{14} \sin \frac{11\pi}{14} \sin \frac{13\pi}{14}$ is equal to

Similar Questions

If $\cos \theta - \sin \theta = \sqrt{2} \sin \theta,$ then $\cos \theta + \sin \theta$ is equal to

If $\alpha$ is a root of $25\cos^2\theta + 5\cos\theta - 12 = 0$ and $\pi/2 < \alpha < \pi$,then $\sin 2\alpha$ is equal to:

If $\theta$ is in the interval $\left(0, \frac{\pi}{2}\right)$ satisfying the equation $\cos 2 \theta \cdot \sec ^4 \theta + \sec ^2 \theta = 0$,then $\sin ^2 \theta =$

For a positive integer $n$,let ${f_n}(\theta ) = \left( {\tan \frac{\theta }{2}} \right)(1 + \sec \theta )(1 + \sec 2\theta )(1 + \sec 4\theta ) \dots (1 + \sec {2^n}\theta ).$ Then

If $a = \frac{x}{y-z}$,$b = \frac{y}{z-x}$,and $c = \frac{z}{x-y}$,where $x, y$,and $z$ are distinct such that $x-y, y-z, z-x \neq 0$,then what is the value of $ab + bc + ca + abc$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module