# Trying to solve the trig equation <msqrt> 3 + 4 cos 2 </

Trying to solve the trig equation $\sqrt{3+4{\mathrm{cos}}^{2}\left(x\right)}=\frac{\mathrm{sin}\left(x\right)}{\sqrt{3}}+3\mathrm{cos}\left(x\right)$
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Leslie Rollins
My solution goes like this
$\left\{\begin{array}{l}3+4{\mathrm{cos}}^{2}\left(x\right)=\frac{{\mathrm{sin}}^{2}\left(x\right)}{3}+\frac{6}{\sqrt{3}}\mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)+9{\mathrm{cos}}^{2}\left(x\right)\\ \frac{\mathrm{sin}\left(x\right)}{\sqrt{3}}+3\mathrm{cos}\left(x\right)\ge 0\end{array}$
$3\left({\mathrm{sin}}^{2}\left(x\right)+{\mathrm{cos}}^{2}\left(x\right)\right)+4{\mathrm{cos}}^{2}\left(x\right)=\frac{{\mathrm{sin}}^{2}\left(x\right)}{3}+\frac{6}{\sqrt{3}}\mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)+9{\mathrm{cos}}^{2}\left(x\right)$
$2{\mathrm{cos}}^{2}\left(x\right)+\frac{{\mathrm{sin}}^{2}\left(x\right)}{3}-3{\mathrm{sin}}^{2}\left(x\right)+\frac{6}{\sqrt{3}}\mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)=0$
I multiply by 3 and divide by ${\mathrm{cos}}^{2}\left(x\right)$:
$8{\mathrm{tan}}^{2}\left(x\right)-6\sqrt{3}\mathrm{tan}\left(x\right)-6=0$
Let $t=\mathrm{tan}\left(x\right)$, then
$4{t}^{2}-3\sqrt{3}t-3=0$
$\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}t=\frac{3\sqrt{3}±\sqrt{\left(3\sqrt{3}{\right)}^{2}-4\cdot 4\left(-3\right)}}{2\cdot 4}=\frac{3\sqrt{3}±5\sqrt{3}}{8}=?$