The value of displaystyle∑ r =010 cos 3 ( r π/3) is equal to (- a / b ) then the value of b is (where g.c.d of (a, b) is 1)

Question

The value of displaystyle∑ r =010 cos 3 ( r π/3) is equal to (- a / b ) then the value of b is (where g.c.d of (a, b) is 1) (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

displaystyle∑ r =010 cos 3 ( r π/3)=(1/4) displaystyle∑ r =010(3 cos ( r π/3)+ cos r π) =(1/4)[3( cos 0+ cos (π/3)+ ldots .+ cos (10 π/3) +(1-1+ ldots . .-1=1)].

Q. The value of $r = 0 \sum 10 cos^{3} \frac{r π}{3}$ is equal to $\frac{- a}{b}$ then the value of $b$ is (where g.c.d of $(a, b)$ is 1 $)$

$r = 0 \sum 10 cos^{3} \frac{r π}{3} = \frac{1}{4} r = 0 \sum 10 (3 cos \frac{r π}{3} + cos r π)$ $= \frac{1}{4} [3 (cos 0 + cos \frac{π}{3} + \dots . + cos \frac{10 π}{3} + (1 - 1 + \dots .. - 1 = 1)]$

Q. The value of r=0∑10​cos33rπ​ is equal to b−a​ then the value of b is (where g.c.d of (a,b) is 1)

r=0∑10​cos33rπ​=41​r=0∑10​(3cos3rπ​+cosrπ) =41​[3(cos0+cos3π​+….+cos310π​+(1−1+…..−1=1)]

Q. The value of $r = 0 \sum 10 cos^{3} \frac{r π}{3}$ is equal to $\frac{- a}{b}$ then the value of $b$ is (where g.c.d of $(a, b)$ is 1 $)$

$r = 0 \sum 10 cos^{3} \frac{r π}{3} = \frac{1}{4} r = 0 \sum 10 (3 cos \frac{r π}{3} + cos r π)$ $= \frac{1}{4} [3 (cos 0 + cos \frac{π}{3} + \dots . + cos \frac{10 π}{3} + (1 - 1 + \dots .. - 1 = 1)]$