Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

$I={\sum }_{r=0}^{10}\frac{1}{4}\left(\cos \pi r+3\cos \frac{\pi r}{3}\right)=\frac{1}{4}\left(I_1+I_2\right)$ $\begin{array}{rlrl}& I_1=\sum _{r=0}^{10}\cos \pi r=1-1+1-\dots -1+1=1& I_2=3\sum _{r=0}^{10}\cos \frac{\pi r}{3}=3\cdot \frac{\cos \left(\frac{10}{2}\cdot \frac{\pi }{3}\right)\sin \frac{11\pi }{6}}{\sin \frac{\pi }{6}}=-\frac{3}{2}& \Rightarrow I=\frac{1}{4}\left(1-\frac{3}{2}\right)=-\frac{1}{8}\end{array}$