Answer:
You don't need to single out [tex]\cos \theta[/tex] to solve this question.
To solve:
[tex]\begin{aligned}3 \cos 4 \theta & = -2\\\\\cos 4 \theta & = -\dfrac{2}{3}\\\\4 \theta & = \cos^{-1}\left( -\dfrac{2}{3\right)}\\\\\implies 4 \theta & =2.30053... \pm 2 \pi n, -2.30053...\pm 2 \pi n\\\\\theta & =\dfrac{2.30053...}{4} \pm \dfrac{\pi n}{2}, -\dfrac{2.30053...}{4}\pm \dfrac{\pi n}{2}\end{aligned}[/tex]
So for the given interval [tex]0\leq \theta \leq 2 \pi[/tex]
[tex]\implies \theta =0.575, 2.146, 3.717, 5.288, 0.996, 2.566, 4.137, 5.708\:\:(\sf 3 \: d.p.)[/tex]
(As confirmed by the attached graph)
[tex]\\ \rm\Rrightarrow 3cos4\theta=-2[/tex]
[tex]\\ \rm\Rrightarrow cos4\theta=\dfrac{-2}{3}[/tex]
[tex]\\ \rm\Rrightarrow 4\theta=cos^{-1}\left(\dfrac{-2}{3}\right)[/tex]
[tex]\\ \rm\Rrightarrow \theta=\pm\dfrac{cos^{-1}\left(\dfrac{-2}{3}\right)}{4}[/tex]
Using scientific calculator
[tex]\\ \rm\Rrightarrow \theta=\pm 32.95257+90°n[/tex]
So as interval is [0,2π] 4 values are there
[tex]\\ \rm\Rrightarrow \theta=32.95257+90°= 122.95277°[/tex]
[tex]\\ \rm\Rrightarrow \theta=-32.95257+90=57.04743°[/tex]
[tex]\\ \rm\Rrightarrow \theta=32.95257+270=302.95257°[/tex]
[tex]\\ \rm\Rrightarrow \theta=32.95257+270=237.04743°[/tex]
