Question

the positive integer value of $n>3$ satisfying the equation
$\frac{1}{\sin \frac{\pi }{n}}=\frac{1}{\sin \frac{2\pi }{n}}+\frac{1}{\sin \frac{3\pi }{n}}$

• A. 5
• B. 6
• C. 7
• D. 8

Answer 1

Answer: (C)

7

$Wehave\frac{1}{\sin \frac{\pi }{n}}=\frac{1}{\sin \frac{2\pi }{n}}+\frac{1}{\sin \frac{3\pi }{n}}\Rightarrow \frac{1}{\sin \frac{\pi }{n}}-\frac{1}{\sin \frac{3\pi }{n}}=\frac{1}{\sin \frac{2\pi }{n}}\Rightarrow \frac{\sin \frac{3\pi }{n}-\sin \frac{\pi }{n}}{\sin \left(\frac{\pi }{n}\right).\sin \frac{3\pi }{n}}=\frac{1}{\sin \frac{2\pi }{n}}\Rightarrow \frac{2\cos \frac{2\pi }{n}\sin \frac{\pi }{n}}{\sin \frac{\pi }{n}\sin \frac{3\pi }{n}}=\frac{1}{\sin \frac{2\pi }{n}}\Rightarrow 2\cos \frac{2\pi }{n}\sin \frac{2\pi }{n}=\sin \frac{3\pi }{n}\Rightarrow \sin \frac{4\pi }{n}=\sin \frac{3\pi }{n}\Rightarrow \frac{4\pi }{n}+\frac{3\pi }{n}=\pi \Rightarrow 7\pi =n\pi$

$\therefore n=7Whichistherequiredanswer.$

Hence,the option $C$ is the correct answer.