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Question Number 193668 by SAMIRA last updated on 17/Jun/23

$$\frac{\sin \left(3\mathrm{x}\right)}{1-\mathrm{x}}-\frac{\cos \left(3\mathrm{x}\right)}{\cos \left(\mathrm{x}\right)}=-2????$$

Answered by Frix last updated on 18/Jun/23

$$\mathrm{Transform}\mathrm{to}$$$$x=-\frac{{4\sin }^{3}x-{4\sin }^{2}x-3\sin x-1}{1+{4\sin }^{2}x}=f\left(x\right)$$$${\mathrm{It}}^{\prime }\mathrm{s}\mathrm{easy}\mathrm{to}\mathrm{show}\mathrm{that}$$$$1-\frac{\sqrt{-9+6\sqrt{3}}}{2}⩽f\left(x\right)⩽1+\frac{\sqrt{-9+6\sqrt{3}}}{2}$$$$\approx .41⩽f\left(x\right)⩽1.59$$$$\Rightarrow$$$$\mathrm{We}\mathrm{find}\mathrm{the}\mathrm{solution}\mathrm{for}.41⩽x⩽1.59$$$$\frac{\sin 3x}{1-x}-\frac{\cos 3x}{\cos x}+2=0$$$$\frac{(3-{4\sin }^{2}x)\sin x}{1-x}+1+{4\sin }^{2}x=0$$$$\mathrm{We}\mathrm{see}{\mathrm{there}}^{\prime }\mathrm{s}\mathrm{a}\mathrm{change}\mathrm{of}\mathrm{sign}\mathrm{around}\mathrm{the}$$$$\mathrm{singularity}\mathrm{at}x=1\Rightarrow \mathrm{we}\mathrm{find}\mathrm{the}\mathrm{solution}$$$$\mathrm{for}1<x⩽1.59$$$$\mathrm{We}\mathrm{can}\mathrm{only}\mathrm{approximate}$$$$\Rightarrow$$$$x\approx 1.02049$$

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