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This question already has an answer here:

Real Analysis Prove that 2/π ≤(sinx)/x ≤ 1 for all |x|≤ π/2 ? Just need the 2/π greater than part.

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marked as duplicate by Clement C., John Doe, Leucippus, Misha Lavrov, Jack D'Aurizio real-analysis Apr 7 '18 at 16:21

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Hint:

On the interval $\bigl[0,\frac\pi2\bigr]$, the sine function is concave. As a consequence, the slopes of the chords joining a point of the curve to the origin are decreasing, and $\frac2\pi$ is the slope of the chord joining the local maximum $\bigl(\frac\pi2,1\bigr)$ to the origin.

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  • $\begingroup$ You mean that in the interval the slope of any point on sine curve (that is cosx ) is greater than the chord with slope 2/π.How are you then getting it in sinx / x form ? Integration of cosx < 2/π ? Sounds correct but I am looking for a solution based on monotonicity of functions. Thank you. $\endgroup$ – DEEPANSHU KAUL PHILIP Apr 7 '18 at 18:13
  • $\begingroup$ The slope of the chord joining the origin to another point (or any fixed point to another point on its right), not the slope of the tangent. $\endgroup$ – Bernard Apr 7 '18 at 18:18
  • $\begingroup$ That is fine but how is that leading up to 2/π ≤ sin(x) / x ? $\endgroup$ – DEEPANSHU KAUL PHILIP Apr 7 '18 at 18:37
  • $\begingroup$ $\sin x/x$ is the slope of the chord joining the point with abscissa $x$ to the origin. It decreases down to its value at $\frac\pi2$. $\endgroup$ – Bernard Apr 7 '18 at 18:40
  • $\begingroup$ Alright got it ! Thank you. $\endgroup$ – DEEPANSHU KAUL PHILIP Apr 7 '18 at 18:47
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We can assume that $$0<x\le \frac{\pi}{2}$$ since $$\frac{\sin(x)}{x}$$ is even. We multiply by $x>0$ and we have to prove that $$\frac{2}{\pi}x\le\sin(x)\le x$$. Defining $$f(x)=x-\sin(x)$$ then we get $$f'(x)=1-\cos(x)>0$$ and let $$g(x)=\sin(x)-\frac{2}{\pi}x$$ then we get $$g'(x)=\cos(x)-\frac{2}{\pi}$$ and $$g''(x)=-\sin(x)<0$$ Can you finish?

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  • $\begingroup$ Sir, I did something similar. Unlike the f(x) function, g(x) is not strictly monotonic. It is changing slope at x= cos–¹(2/π). Also, please explain a bit more on how to get 2/π xsinx ≤ x ? $\endgroup$ – DEEPANSHU KAUL PHILIP Apr 7 '18 at 18:31

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