Question

Find the sum of the three smallest positive values of
`$\theta$` such that
`$4\cos^2(2\theta-\pi) =3$`.

Answer 1

`\boxed{\boxed{(13\pi)/(12)}}`

Explanation:

`4\cos^2(2\theta-\pi)=3\qquad\text{divide both sides by 4}\\\\\cos^2(2\theta-\pi)=(3)/(4)\to\cos(2\theta-\pi)=\pm\sqrt{(3)/(4)}\\\\\cos(2\theta-\pi)=-(\sqrt3)/(2)\ \vee\ \cos(2\theta-\pi)=(\sqrt3)/(2)`

`2\theta-\pi=(5\pi)/(6)+2k\pi\ \vee\ 2\theta-\pi=-(5\pi)/(6)+2k\pi\qquad k\in\mathbb{Z}\\\\2\theta-\pi=(\pi)/(6)+2k\pi\ \vee\ 2\theta-\pi=-(\pi)/(6)+2k\pi\qquad k\in\mathbb{Z}\\\\\text{add}\ \pi\ \text{to both sides}\\\\2\theta=(11\pi)/(6)+2k\pi\ \vee\ 2\theta=(\pi)/(6)+2k\pi\\\\2\theta=(7\pi)/(6)+2k\pi\ \vee\ 2\theta=(5\pi)/(6)+2k\pi\\\\\text{divide both sides by 2}`

`\theta=(11\pi)/(12)+k\pi\ \vee\ \theta=(\pi)/(12)+k\pi\ \vee\ \theta=(7\pi)/(12)+k\pi\ \vee\ \theta=(5\pi)/(12)+k\pi\\\\\qquad k\in\mathbb{Z}`

`\text{The three smallest positive values of}\ \theta:\\\\(\pi)/(12),\ (5\pi)/(12),\ (7\pi)/(12)\\\\\text{The sum:}\\\\S=(\pi)/(12)+(5\pi)/(12)+(7\pi)/(12)=(\pi+5\pi+7\pi)/(12)=(13\pi)/(12)`