Solve the equation on the interval 0 lessthanorequalto theta

Solve the equation on the interval 0 lessthanorequalto theta

Solution

9cos(2)=-27cos-18
dividing both sides by 9
cos(2)=-3cos-2
(2cos² - 1)=-3cos-2
adding +3cos+2 on both sides
(2cos² - 1)+3cos+2=-3cos-2+3cos+2
so, we will be left with
2cos² +3cos+1=0
2cos² +2cos+1cos+1=0
let us take 2cos common from first two terms and 1 common from last two terms,
2cos(cos+1)+1(cos+1)=0
now let us take (cos+1) common
(cos+1)(2cos+1)=0
now we can equate each factor equal to 0
2cos+1=0
2cos=-1
dividing both sides by 2
(2cos)/2=-1/2
cos=-1/2
and
(cos+1)=0
cos=-1
so now cos=-1 and -1/2 also we are given the interval here
now see cos is negative in 2nd and 3rd quadrant
so value of are 120degrees, 240degrees and 180degrees
in radians: 2pi/3,4pi/3,pi