Question

Find the value of cos Ө + cos(2 Π/3+ Ө )+ cos (2 Π/3- Ө )

Answer 1

$$\begin{gathered} \mathrm{To}\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}\cos \mathrm{\theta }+\cos \left(\frac{2\mathrm{\pi }}{3}+\mathrm{\theta }\right)+\cos \left(\frac{2\mathrm{\pi }}{3}-\mathrm{\theta }\right) \\ =\cos \mathrm{\theta }+2\cos \left(\frac{2\mathrm{\pi }}{3}\right)\mathrm{cos\theta }\left(\mathrm{since}\cos \left(\mathrm{A}\right)+\cos \left(\mathrm{B}\right)=2\cos \left(\frac{\mathrm{A}+\mathrm{B}}{2}\right)\cos \left(\frac{\mathrm{A}-\mathrm{B}}{2}\right)\right) \\ =\cos \mathrm{\theta }+2\cos \left(\mathrm{\pi }-\frac{\mathrm{\pi }}{3}\right)\cos \mathrm{\theta } \\ =\cos \mathrm{\theta }-2\cos \left(\frac{\mathrm{\pi }}{3}\right)\cos \mathrm{\theta }\left(\mathrm{since}\cos \left(\mathrm{\pi }-\mathrm{x}\right)=-\cos \mathrm{x}\right) \\ =\cos \mathrm{\theta }-2\left(\frac{1}{2}\right)\cos \mathrm{\theta }\left(\mathrm{since}\cos \left(\frac{\mathrm{\pi }}{3}\right)=\frac{1}{2}\right) \\ =\cos \mathrm{\theta }-\cos \mathrm{\theta } \\ =0 \\ \mathrm{So},\mathrm{value}\mathrm{of}\cos \mathrm{\theta }+\cos \left(\frac{2\mathrm{\pi }}{3}+\mathrm{\theta }\right)+\cos \left(\frac{2\mathrm{\pi }}{3}-\mathrm{\theta }\right)\mathrm{is}0. \end{gathered}$$