Question

The value of $\cos \left(\frac{2\mathrm{\pi }}{7}\right)+\cos \left(\frac{4\mathrm{\pi }}{7}\right)+\cos \left(\frac{6\mathrm{\pi }}{7}\right)=-\frac{1}{\mathrm{a}}$. Find $\mathrm{a}$.

Answer 1

Determine the value of $\mathrm{a}$ in $\cos \left(\frac{2\mathrm{\pi }}{7}\right)+\cos \left(\frac{4\mathrm{\pi }}{7}\right)+\cos \left(\frac{6\mathrm{\pi }}{7}\right)=-\frac{1}{\mathrm{a}}$.

$$\begin{gathered} \cos \left(\frac{2\mathrm{\pi }}{7}\right)+\cos \left(\frac{4\mathrm{\pi }}{7}\right)+\cos \left(\frac{6\mathrm{\pi }}{7}\right)=-\frac{1}{a} \\ \Rightarrow 2\cos \frac{3\mathrm{\pi }}{7}\cos \frac{\mathrm{\pi }}{7}+\cos \frac{6\mathrm{\pi }}{7}=-\frac{1}{a} \\ \Rightarrow 2\cos \frac{3\mathrm{\pi }}{7}\cos \frac{\mathrm{\pi }}{7}-\cos \frac{\mathrm{\pi }}{7}=-\frac{1}{a} \\ \Rightarrow \cos \frac{\mathrm{\pi }}{7}\left(2\cos \frac{3\mathrm{\pi }}{7}-1\right)=-\frac{1}{a} \\ \Rightarrow \cos \frac{\mathrm{\pi }}{7}\frac{\left(2\cos \frac{3\mathrm{\pi }}{7}-1\right)\sin {\displaystyle \frac{3\mathrm{\pi }}{7}}}{\sin \frac{3\mathrm{\pi }}{7}}=-\frac{1}{a} \\ \Rightarrow \frac{\cos \frac{\mathrm{\pi }}{7}\left(2\cos \frac{3\mathrm{\pi }}{7}\cos \frac{3\mathrm{\pi }}{7}-\sin \frac{3\mathrm{\pi }}{7}\right)}{\sin \frac{3\mathrm{\pi }}{7}}=-\frac{1}{a} \\ \Rightarrow \frac{\cos \frac{\mathrm{\pi }}{7}\left(\sin \frac{6\mathrm{\pi }}{7}-\sin \frac{3\mathrm{\pi }}{7}\right)}{\sin \frac{3\mathrm{\pi }}{7}}=-\frac{1}{a} \\ \Rightarrow \frac{\cos \frac{\mathrm{\pi }}{7}\left(2\sin \frac{3\mathrm{\pi }}{14}\cos \frac{9\mathrm{\pi }}{14}\right)}{2\sin \frac{3\mathrm{\pi }}{14}\cos \frac{3\mathrm{\pi }}{14}}=-\frac{1}{a} \\ \Rightarrow \frac{\cos \frac{\mathrm{\pi }}{7}\cos \frac{9\mathrm{\pi }}{14}}{\cos \frac{3\mathrm{\pi }}{14}}=-\frac{1}{a} \\ \Rightarrow \frac{1}{2}\frac{\left(\cos \frac{11\mathrm{\pi }}{14}+\cos \frac{7\mathrm{\pi }}{14}\right)}{\cos \frac{3\mathrm{\pi }}{14}}=-\frac{1}{a} \\ \Rightarrow \frac{1}{2}\left(\frac{-\cos \frac{3\mathrm{\pi }}{14}+0}{\cos \frac{3\mathrm{\pi }}{14}}\right)=-\frac{1}{a} \\ \Rightarrow -\frac{1}{2}=-\frac{1}{a} \\ \Rightarrow a=2 \end{gathered}$$

Hence, the value of $\mathrm{a}$ is $2$.