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

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Answer 1 · 2018-05-07T03:24:09

$(13pi)/12$

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]}