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Question Number 55971 by bshahid010@gmail.com last updated on 07/Mar/19

Commented by tanmay.chaudhury50@gmail.com last updated on 07/Mar/19

$$fromgraphitisseenthat\mid sinx\mid and\frac{\sqrt{3}}{2}\mid x\mid meetat$$$$(0,0)fornosolutionashouldbegreaterthanzero$$$$\mid sinx\mid =\frac{\sqrt{3}}{2}\mid x\mid meetat(0,0)sox=0isasolutiin$$$$but\mid sinx\mid =\frac{\sqrt{3}}{2}\mid x\mid +ahavenosolutionwhen$$$$a>0$$$$$$

Commented by tanmay.chaudhury50@gmail.com last updated on 07/Mar/19

Answered by mr W last updated on 07/Mar/19

Commented by mr W last updated on 07/Mar/19

$$duetosymmetryweonlyneedtolook$$$$atx⩾0.$$$$y=sin\left(x\right)$$$$y^{\prime }=\cos \left(x\right)=\frac{\sqrt{3}}{2}$$$$\Rightarrow x=\frac{\pi }{6}$$$$i.e.atx=\frac{\pi }{6},LHS=RHS$$$$\sin \frac{\pi }{6}=\frac{\sqrt{3}}{2}\times \frac{\pi }{6}+a$$$$\Rightarrow a=\frac{6-\sqrt{3}\pi }{12}\approx 0.04655$$$$i.e.ifa⩽\frac{6-\sqrt{3}\pi }{12}\approx 0.04655,$$$$thereisalwaysatleastonesolution$$$$for\mid \sin x\mid =\frac{\sqrt{3}}{2}\mid x\mid +a.$$$$andifa>\frac{6-\sqrt{3}\pi }{12}\approx 0.04655thereisno$$$$solutionfor\mid \sin x\mid =\frac{\sqrt{3}}{2}\mid x\mid +a.$$

Commented by tanmay.chaudhury50@gmail.com last updated on 07/Mar/19

$$bahdarunexcellent...$$

Commented by mr W last updated on 07/Mar/19

$$\mathbb{T}\mathit{o}\mathit{k}\mathit{e}\mathit{d}\mathit{h}\mathit{o}\mathit{n}\mathit{n}\mathit{o}\mathit{b}\mathit{a}\mathit{d}!$$

Commented by mr W last updated on 07/Mar/19

$$hope{it}^{\prime }scorrectlywritten.$$

Commented by tanmay.chaudhury50@gmail.com last updated on 07/Mar/19

$$namki...nischoibangali...amibarrackporear...service$$$$inrailway...nagpurethaki...$$

Commented by bshahid010@gmail.com last updated on 07/Mar/19

$$\mathrm{thank}\mathrm{u}$$

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