solve the equation on the interval 0 2 9 cos2 theta 27cos t

solve the equation on the interval
0 2
9 cos(2 theta) = -27cos theta - 18

Solution

9cos 2theta=27cos theta-18

9(2cos²theta-1)=27cos theta-18

18cos²theta -9 -27 cos theta+18=0

18cos²theta -27cos theta +9=0

2cos²theta -3cos theta +1=0

2cos²theta - 2cos theta - cos theta +1=0

2cos theta(cos theta-1)-1(cos theta-1)=0

(2cos theta-1)(cos theta -1)=0

cos theta=1/2    or      cos theta=1

Taking cos^-1 on both sides

theta=cos^-11/2          or      theta=cos^-11

theta=pi/3, 5pi/3         or   theta= 0