Question

-4cos²(x) - 7cos(x) = 3

Answer 1

`-4\cos^2(x)-7\cos(x)=3\hspace{5em}\stackrel{\textit{let's for a few seconds make}}{\cos(x)=c} \\\\\\ -4c^2-7c=3\implies 0=4c^2+7c+3\implies 0=(4c+3)(c+1) \\\\[-0.35em] ~\dotfill\\\\ 0=4\cos(x)+3\implies -3=4\cos(x)\implies \cfrac{-3}{4}=\cos(x) \\\\\\ \cos^(-1)\left( \cfrac{-3}{4} \right)=x\implies x\approx \begin{cases} \stackrel{ II~Quadrant }{2.4189~rad}\\\\ \stackrel{ III~Quadrant }{3.8643~rad} \end{cases} \\\\[-0.35em] ~\dotfill\\\\ 0=\cos(x)+1\implies -1=\cos(x)\implies \cos^(-1)(-1)=x\implies \pi =x`

bear in mind that the inverse cosine function has a range of from 0 to π, so any values it spits out it'll be on those Quadrants, to get the III Quadrant value, we simply use the reference angle from the II Quadrant angle, keeping in mind that the cosine is negative on the II and III Quadrants only.