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Question Number 56681 by Joel578 last updated on 21/Mar/19

$$\mathrm{Prove}\mathrm{that}$$ $$\sin \mid x\mid ⩽\mid x\mid ⩽\tan \mid x\mid \mathrm{for}\mid x\mid <\frac{\pi }{2}$$

Commented byAbdo msup. last updated on 22/Mar/19

$$let\mid x\mid =tletprovethatsint⩽t⩽tantfor0⩽t<\frac{\pi }{2}$$ $$letW\left(x\right)=t-sint\Rightarrow W^{,}\left(t\right)=1-cost⩾0$$ $$t0\frac{\pi }{2}$$ $$W^{{}^{\prime }}+$$ $$W0incr1\Rightarrow W\left(t\right)⩾0\Rightarrow sint⩽t$$ $$let\phi \left(t\right)=tant-t\Rightarrow {\phi }^{{}^{\prime }}\left(t\right)=1+{tan}^{2}t-1={tan}^{2}t⩾0\Rightarrow$$ $$t0\frac{\pi }{2}$$ $${\phi }^{{}^{\prime }}+$$ $$\phi 0+\mathrm{\infty }\Rightarrow \phi \left(t\right)⩾0\Rightarrow t⩽tant$$ $$sotheresultisproved.$$

Answered by ajfour last updated on 21/Mar/19

$$\mathrm{let}\mid \mathrm{x}\mid =\mathrm{x}$$ $$\cos \mathrm{x}⩽1⩽1+\tan {}^2\mathrm{x}$$ $$\frac{\mathrm{d}\left(\sin \mathrm{x}\right)}{\mathrm{dx}}⩽\frac{\mathrm{d}\left(\mathrm{x}\right)}{\mathrm{dx}}⩽\frac{\mathrm{d}\left(\tan \mathrm{x}\right)}{\mathrm{dx}}$$ $$\mathrm{and}\mathrm{at}\mathrm{x}=0,\sin \mathrm{x}=\mathrm{x}=\tan \mathrm{x};$$ $$\mathrm{hence}\sin \mathrm{x}⩽\mathrm{x}⩽\tan \mathrm{x}.$$

Answered by mr W last updated on 21/Mar/19

Commented bymr W last updated on 21/Mar/19

$$OA=1$$ $$\mathrm{\angle }AOB=x$$ $$\stackrel{⌢}{AB}=x$$ $$AC=\sin x$$ $$AD=\tan x$$ $$AC<\stackrel{⌢}{AB}<AD$$ $$\Rightarrow \sin x<x<\tan x$$

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