Question

$\cos \left(\frac{2\mathrm{\pi }}{7}\right)+\cos \left(\frac{4\mathrm{\pi }}{7}\right)+\cos \left(\frac{6\mathrm{\pi }}{7}\right)=?$

• A. Is equal to zero
• B. Lies between$0\&3$
• C. Is a negative number
• D. Lies between $3\&6$

Answer 1

The correct option is C

Is a negative number

Finding the value for $\mathit{c}\mathit{o}\mathit{s}\left(\frac{2\pi }{7}\right)\mathbf{+}\mathit{c}\mathit{o}\mathit{s}\left(\frac{4\pi }{7}\right)\mathbf{+}\mathit{c}\mathit{o}\mathit{s}\left(\frac{6\pi }{7}\right)\mathbf{:}$

given,

Multiply and divide by $2\sin \left(\frac{\mathrm{\pi }}{7}\right)$

$\begin{array}{rcl}& & \cos \left(\frac{2\mathrm{\pi }}{7}\right)+\cos \left(\frac{4\mathrm{\pi }}{7}\right)+\cos \left(\frac{6\mathrm{\pi }}{7}\right)\\ & =& \frac{2\sin {\displaystyle \left(\frac{\mathrm{\pi }}{7}\right)}\cos \left(\frac{2\mathrm{\pi }}{7}\right)+2\sin \left(\frac{\mathrm{\pi }}{7}\right)\cos \left(\frac{4\mathrm{\pi }}{7}\right)+2\sin \left(\frac{\mathrm{\pi }}{7}\right)\cos \left(\frac{6\mathrm{\pi }}{7}\right)}{2\sin \left(\frac{\mathrm{\pi }}{7}\right)}\end{array}$

Using formula $\mathbf{2}\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{y}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathit{x}\mathbf{+}\mathit{y}\mathbf{)}\mathbf{+}\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathit{x}\mathbf{–}\mathit{y}\mathbf{)}$

$\begin{array}{rcl}& & \frac{2\sin \left(\frac{\mathrm{\pi }}{7}\right)\cos \left(\frac{2\mathrm{\pi }}{7}\right)+2\sin \left(\frac{\mathrm{\pi }}{7}\right)\cos \left(\frac{4\mathrm{\pi }}{7}\right)+2\sin \left(\frac{\mathrm{\pi }}{7}\right)\cos \left(\frac{6\mathrm{\pi }}{7}\right)}{2\sin \left({\displaystyle \frac{\mathrm{\pi }}{7}}\right)}\\ & =& \frac{\sin \left(\frac{3\mathrm{\pi }}{7}\right)+\sin {\displaystyle \left(\frac{-\mathrm{\pi }}{7}\right)}+\sin \left(\frac{5\mathrm{\pi }}{7}\right)+\sin \left(\frac{-3\mathrm{\pi }}{7}\right)+\mathrm{sin\pi }+\sin \left({\displaystyle \frac{-5\mathrm{\pi }}{7}}\right)}{2\sin \left(\frac{\mathrm{\pi }}{7}\right)}\\ & =& \frac{\sin \left({\displaystyle \frac{3\mathrm{\pi }}{7}}\right)-\sin \left({\displaystyle \frac{\mathrm{\pi }}{7}}\right)+\sin \left({\displaystyle \frac{5\mathrm{\pi }}{7}}\right)-\sin \left({\displaystyle \frac{3\mathrm{\pi }}{7}}\right)+\mathrm{sin\pi }-\sin \left({\displaystyle \frac{5\mathrm{\pi }}{7}}\right)}{2\sin \left({\displaystyle \frac{\mathrm{\pi }}{7}}\right)}\\ & =& \frac{-\sin \left({\displaystyle \frac{\mathrm{\pi }}{7}}\right)}{2\sin \left({\displaystyle \frac{\mathrm{\pi }}{7}}\right)}\\ & =& \frac{-1}{2}\end{array}$

Hence, correct option is $\left(C\right)$