Archimedes used the fact that

$\sin(x)< x< \tan(x)$ when $0< x < \cfrac{\pi}2$

to prove that the perimeter of a polygon inscribed in a circle is less than the circumference of the circle. Similarly, he used this to show that the perimeter of a polygon circumscribed about a circle exceeds the circumference of the circle. I want to get some idea as to how Archimedes proved the above result.

  • $\begingroup$ Maybe just plot the graph? $\endgroup$ – D F Feb 7 '18 at 5:52
  • $\begingroup$ Look at $f(x)=\sin x-x$ and $g(x)=x-\tan x$. Using calculus it's easy to see these inequalities are true: $f'(x) \lt 0$ and $g'(x) \lt 0$ for $x\lt \frac {\pi}2$, with $f(0)=g(0)=0$... $\endgroup$ – Chris Custer Feb 7 '18 at 6:09
  • $\begingroup$ Archimedes had techniques equivalent to modern day calculus... but i have no idea how he established these inequalities... Remember he was one of the four greatest ever. (Euler, Gauss and Newton being the other three...) $\endgroup$ – Chris Custer Feb 7 '18 at 6:15
  • $\begingroup$ See my answer here for example. $\endgroup$ – dxiv Feb 7 '18 at 6:54
  • $\begingroup$ @ChrisCuster - "Archimedes had techniques equivalent to modern day calculus", really? care to back that statement up? $\endgroup$ – nbubis Feb 7 '18 at 10:49

Consider the following triangles in the unit circle: enter image description here

We know the following: $\overline{XY} =\overline{XZ} = 1$, $\overline{XV} = \cos\theta$, $\overline{ZV} = \sin\theta$, $\overline{WY} = \tan\theta$. Now, the area of the triangle $\Delta XZY$ is obviously $\frac{1}{2}\sin\theta$. The area of the sector of the unit circle containing $\Delta XZY$ is $\frac{1}{2}\theta$. Finally, the area of $\Delta XWY$ is $\frac{1}{2}\tan\theta$. Since $\Delta XWY$ contains the sector, which contains $\Delta XZY$, we have: $$\frac{1}{2}\sin\theta < \frac{1}{2}\theta < \frac{1}{2}\tan\theta$$ It is easy to go from this to $$\sin\theta < \theta < \tan\theta$$

| cite | improve this answer | |
  • $\begingroup$ Thanks! May I ask you how did you produce this image? I'm trying to find a good way to produce mathematical diagrams. $\endgroup$ – k13 Feb 7 '18 at 16:29
  • 1
    $\begingroup$ I used the TikZ package for LaTeX $\endgroup$ – bames Feb 7 '18 at 18:15

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.