Question

Find the sum of the three smallest positive values of θ such that 4 cos^2(20-π)=3?

Answer 1

`(13pi)/12`

Explanation:

I will assume it says:

`4 cos^2(2theta-π)=3`

Let `u` be `2theta-pi`

`4cos^2u=3`

`cos^2u= 3/4`

`cosu= +-sqrt3/2`

Now I will list all my solutions from `(-pi, pi)` instead of `(0, 2pi)` for a reason I will discuss later:

`u= -(5pi)/6, -(pi)/6, pi/6, (5pi)/6`

Now replace `u` with `2theta-pi`:

The reason why I listed solutions from `-pi` is because I will be adding `pi` here:
`2theta-pi=-(5pi)/6, -(pi)/6, pi/6, (5pi)/6`

`2theta= pi/6, (5pi)/6, (7pi)/6, (11pi)/6`

`theta= pi/12, (5pi)/12, (7pi)/12, (11pi)/12`

Sum of the three smallest positive solutions:
`pi/12+(5pi)/12+(7pi)/12= (13pi)/12`

graph{4(cos(2x-pi))^2-3 [-10, 10, -5, 5]}