# Find the maximum angle X in the range 0^{\circ} \leq

Find the maximum angle X in the range ${0}^{\circ }\le x\le {360}^{\circ }$ which satisfies the equation ${\mathrm{cos}}^{2}\left(2x\right)+\sqrt{3}\mathrm{sin}\left(2x\right)-\frac{74}{=}0$
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basgrwthej
Note you have ${\mathrm{cos}}^{2}2x=1-{\mathrm{sin}}^{2}2x$; with $\mathrm{sin}2x=t$ you get
$1-{t}^{2}+t\sqrt{3}-\frac{74}{=}0$
so the equation can be rewritten
$4{t}^{2}-4t\sqrt{3}+3=0$
This is actually ${\left(2t-\sqrt{3}\right)}^{2}=0$, so the only root is $t=\frac{\sqrt{3}}{2}$. Hence . Hence

If you prefer degrees,

It shouldn't be difficult to find the maximum solution.