Question

How to find the general equation of `2cos2theta=4costheta-3`?

Answer 1

`theta = pi/3+2npi` where n is an integer

Explanation:

`2cos(2theta)=4costheta-3`

`2(cos^2theta-sin^2theta)=4costheta-3`

`2(2cos^2theta-1)=4costheta-3`

`4cos^2theta-2=4costheta-3`

`4cos^2theta-4costheta+1=0`

`(2costheta-1)^2=0`

`2costheta-1=0`

`costheta=1/2`

`theta= cos^(-1)(1/2)`

`theta = pi/3+2npi` where n is an integer

Answer 2

The general solution is `θ = 2 n pi ± 60`, where `n in Z`

Explanation:

Note: General solution instead of general equation is required

`2 cos 2 theta = 4 cos theta -3` or

`2 (2 cos^2 theta -1) = 4 cos theta -3` or

`4 cos^2 theta -2 = 4 cos theta -3` or

`4 cos^2 theta -2 -4 cos theta +3 =0` or

`4 cos^2 theta -4 cos theta +1 =0` or

`(2 cos theta -1)^2=0 :. 2 cos theta -1=0` or

`2 cos theta =1 or cos theta = 1/2 ; cos 60 =1/2`

and `cos (-60)=1/2 :. alpha= +-60^0`

The general solution is `θ =2 n pi ± 60`, where `n in Z` [Ans]