Question

Assertion :$\sin \frac{2\pi }{7}+\sin \frac{4\pi }{7}+\sin \frac{8\pi }{7}=\frac{\sqrt{7}}{2}$ Reason: $\cos \frac{2\pi }{7}+i\sin \frac{2\pi }{7}$ is the complex 7th root of unity

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertionfalse but Reason is true

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Let $S=\sin \frac{2\pi }{7}+\sin \frac{4\pi }{7}+\sin \frac{8\pi }{7}andC=\cos \frac{2\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{8\pi }{7}$
$\therefore C+S=(\cos \frac{2\pi }{7}+\sin \frac{2\pi }{7})+(\cos \frac{4\pi }{7}+\sin \frac{4\pi }{7})+(\cos \frac{8\pi }{7}+\sin \frac{8\pi }{7})$
$C+iS=\alpha +{\alpha }^2+{\alpha }^4$ .....(i)
and $C-iS={\alpha }^{-1}+{\alpha }^{-2}+{\alpha }^{-4}$ ....( ii )
$\therefore 2C=\alpha +{\alpha }^2+{\alpha }^4+{\alpha }^{-1}+{\alpha }^{-2}+{\alpha }^{-4}$
$=\alpha +{\alpha }^2+{\alpha }^4+{\alpha }^6+{\alpha }^5+{\alpha }^3(\because {\alpha }^7=1)$
$=\alpha +{\alpha }^2+{\alpha }^3+{\alpha }^4+{\alpha }^5+{\alpha }^6+{\alpha }^7-{\alpha }^7$
$2C=0-{\alpha }^7=-1$
$C=-\frac{1}{2}$
And
$C^2+S^2=(\alpha +{\alpha }^2+{\alpha }^4)({\alpha }^3+{\alpha }^5+{\alpha }^6)=2{\alpha }^7+{\alpha }^1+{\alpha }^2+{\alpha }^3+{\alpha }^4+{\alpha }^5+{\alpha }^6+{\alpha }^7\left(\begin{array}{c}{\alpha }^8={\alpha }^2\\ {\alpha }^9={\alpha }^3\\ {\alpha }^{10}=\alpha \end{array}\right)=2{\alpha }^7+0=2(\because {\alpha }^7=1)\therefore S^2=2-\frac{1}{4}=\frac{7}{4},S=\frac{\sqrt{7}}{2}$