Question

The value of $\sum _{r=0}^{10}$ $co{s}^3$ $\frac{\pi r}{3}$ is equal to

• A. -9/2
• B. -7/2
• C. -9/8
• D. -1/8

Answer 1

The correct option is D

-1/8

I = $\sum _{r=0}^{10}$ $\frac{1}{4}$(cos $\frac{3\pi r}{3}$ + 3cos $\frac{\pi r}{3}$)

= $\sum _{r=0}^{10}$ $\frac{1}{4}$(cos$\pi$r + 3cos $\frac{\pi r}{3}$)

= $\frac{1}{4}$$(I_1+I_2)$

∴ $I_1$ = $\sum _{r=0}^{10}$ cos$\pi$r = 1 - 1 + 1 - 1 +.........-1 + 1 = 1

$I_2$ = $\sum _{r=0}^{10}$ cos$\frac{\pi r}{3}$ = $\frac{cos(\frac{10}{2}.\frac{\pi }{3}).sin\frac{11\pi }{6}}{sin\frac{\pi }{6}}$ = - $\frac{1}{2}$

$I$ = $\frac{1}{4}$(1 - $\frac{3}{2}$) = - $\frac{1}{8}$