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One of my friends gave me this problem. Initially I thought that this would be easy but i tried this for two days unable to do this in a proper manner.

enter image description here

There is a circle with an angle A located at the center of the circle

Prove that: $\sin A < A < \tan A$

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    Include what you've tried so far, please. – 2017-01-30
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    $\tan(3\pi/4) < 0 < 3\pi/4$ so apparently you are missing assumptions. – 2017-01-30
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    Maybe superb jhon means :$A$ is an acute angel – 2017-01-30
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    Probably! superb jhon should *tell us*. – 2017-01-30
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    we need A to be a positive acute angle for this. (A=0 fails) – 2017-01-30
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    See the many answers to [How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?](http://math.stackexchange.com/questions/75130/how-to-prove-that-lim-limits-x-to0-frac-sin-xx-1). – 2017-01-30
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    @Khosrotash I would fear the wings of an acute angel. – 2017-01-30

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enter image description here The area of $\triangle ABC$ is $\frac{1}{2}\sin(x)$. The area of the colored wedge is $\frac{1}{2}x$, and the area of $\triangle ABD$ is $\frac{1}{2}\tan(x)$. By inclusion, we get $$ \frac{1}{2}\tan(x)\ge\frac{1}{2}x\ge\frac{1}{2}\sin(x)\tag{1} $$

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    What s/w do you use to create that picture? – 2017-01-30
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    Combining from paint and desmos.com .... – 2017-01-30
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On $(0,\pi/2)$ we have $\cos t < 1 < \sec^2 t.$ Integrating each term from $t=0$ to $t=A$ gives your desired inequality as long as $0

Of course this proof is usually done via a diagram; see a calc book.

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The shortest way between two points is the straight line. This establishes $$\sin \dfrac A2<\dfrac A2.$$

Then the area of the triangle formed with the tangent to the circle is larger than the area of the sector, hence

$$\frac A2<\tan\frac A2.$$

Replace $\dfrac A2$ by $A$.

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    Could you please be a slight bit more explicit about your reasoning? It is really hard to follow entirely. I see what you mean but I think this post could be improved by a bit more exposition (writing) about why the length between two points correlates to sin and the angle and why tangent correlates to area, etc. – 2017-01-31