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I'm trying to show the inequality $$\frac{\sin(x)}{x}>\cos(x)$$ by for $0<x<\pi$ using the Mean Value Theorem, but I don't know how to start. I can show that $\sin(x)<x$, but I can't see how I can use it. I just need some help to get started.

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  • $\begingroup$ Hint: If you pick the two points $x$ and $0$, $\frac{\sin(x)}{x}$ is the secant between them. $\endgroup$ – Michael Burr Aug 29 '15 at 13:40
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This is not true in general.

For an interval of clear counterexamples, consider that for $x\in(\frac32\pi,2\pi)$ we have $$ \frac{\sin x}{x} < 0 < \cos x$$


Update after the question was amended to specify $0<x<\pi$:

The mean value theorem says that $\frac{\sin x}{x} = \sin'(\alpha)$ for some $\alpha\in(0,x)$. We have $\sin'(\alpha)=\cos\alpha$ so what you need to show is merely that $\cos \alpha > \cos x$. Hopefully you already know that the cosine decreases monotonically between $0$ and $\pi$...

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  • $\begingroup$ I'm sorry...it should be for 0<x<pi $\endgroup$ – netwon1227 Aug 29 '15 at 13:34
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Consider the function $f(x)=\sin(x)$. Fix a point $y$ between $0$ and $\pi$. By applying the mean value theorem to the points $x=0$ and $x=y$, you know that for some point $z$ between $0$ and $y$, $$ \cos(z)=f'(z)=\frac{f(y)-f(0)}{y-0}=\frac{\sin(y)}{y}. $$ Since $\cos(x)$ is a decreasing function on the interval $0$ to $\pi$ and $y>z$, it follows that $\cos(y)<\cos(z)$. Therefore, $$ \cos(y)<\cos(z)<\frac{\sin(y)}{y}. $$

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Since $\cos$ is strictly decreasing on $[0,\pi]$ one has $${\sin x\over x}=\int_0^1\cos(t\>x)\>dt>\cos x\qquad(0<x\leq\pi)\ .$$

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Wait, if you already know how to show $\sin(x) > x$, then you just need to observe that $\cos(x) \leq 1$ for all $x$ to get what you want.

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  • $\begingroup$ But the OP asked for a way using MVT. $\endgroup$ – Michael Burr Aug 29 '15 at 13:59
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    $\begingroup$ He'd better not be able to show $\sin x > x$ on $(0,\pi)$, because that's not true! $\endgroup$ – Henning Makholm Aug 29 '15 at 14:17
  • $\begingroup$ aha, whoops! will delete this answer in a bit. $\endgroup$ – hunter Aug 29 '15 at 15:53

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