The sum of all the solution of the equation `costhetacos(pi/3+theta)cos(pi/3-theta)=1/4theta in [0,6pi]` `15pi` (b) `30pi` (c) `(100pi)/3` (d) none of these

Options

A. `15 pi`
B. `30 pi`
C. `(100 pi)/3`
D. none of these

Answer

B

Explanation

`2 cos theta [cos 120^(@)+cos 2 theta]=1`
or `2 cos theta (-1/2 + 2 cos^(2) theta -1)=1`
or `4 cos^(3) theta -3 cos theta -1=0`
or `cos 3 theta =1=cos 0`
or `3 theta = 2n pi or theta =(2n pi)/3, n in Z`
Given the values so that 2n does not exceed 18.
`:. n=0, 1, 2, 3, ..., 9`
Hence, the sum `=(2pi)/3 sum_(1)^(9) n =(2pi)/3 xx (9(9+1))/2=30 pi`.