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Yesterday I've noticed some relationships with cosine and square root. Anything interesting about it?

I was trying to find the smallest width on an hexagon with radius $1.0$ and I noticed that I could get it both by $\sqrt 3$ and the or two times the cosine of $30^\circ$.

enter image description here

I wondered if there would be more matches and so I notice these even though might look silly...

$2 \times \cos 90^\circ = \sqrt 0$

$2 \times \cos 60^\circ = \sqrt 1$

$2 \times \cos 45^\circ = \sqrt 2$

$2 \times \cos 30^\circ = \sqrt 3$

$2 \times \cos 0^\circ = \sqrt 4$

Is there anything interesting to know about these?

Thanks

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    $\begingroup$ Well, it doesn't look silly at all. It's a well known mnemonic of some important values of $\cos$ and $\sin$ as well. $\endgroup$ – Michael Hoppe Jun 24 '17 at 14:29
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    $\begingroup$ You might notice that the pattern is not as regular as it first seems. The angle difference between $\sqrt 0$ and $\sqrt 1$ and between $\sqrt 3$ and $\sqrt 4$ is $30^\circ$, but the angle differences between $\sqrt 1, \sqrt 2,\sqrt 3$ are $15^\circ$. $\endgroup$ – Paul Sinclair Jun 24 '17 at 16:50
  • $\begingroup$ @PaulSinclair You can see here the graph of the function $f(x) = (sin(x)*2)**2) i.imgur.com/5qrACzH.png $\endgroup$ – probiner Jun 24 '17 at 22:16
  • $\begingroup$ Also: i.imgur.com/GLBPzVN.png $\endgroup$ – probiner Jun 24 '17 at 22:33
  • $\begingroup$ @probiner - I'm sorry, but what was the point of posting those pictures? I can assure that I am quite well aware of what the graphs of trig function (and their squares) look like. $\endgroup$ – Paul Sinclair Jun 25 '17 at 17:16
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Your list is sometimes given in trig books as a mnemonic device. Looking a little deeper the pattern that the angles in a regular $n$-gon give algebraic values for the $\cos$ function is showing itself, though the pattern of simple surds does not continue, e.g. $\cos (\frac{2\pi}{5})=\frac{1}{4}(\sqrt{5}-1)$

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  • $\begingroup$ Did you end your answer? Looks cut. Someone mentioned this to me: purplemath.com/modules/specang.htm $\endgroup$ – probiner Jun 24 '17 at 15:01
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    $\begingroup$ @probiner thanks for pointing out the dangling conjunction. I was thinking about mentioning DeMoivre's Theorem but got interrupted. $\endgroup$ – sharding4 Jun 24 '17 at 15:40
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This rules we used in high school with fingers do apply for both sine and cosine see the pictures

enter image description here

See also the following

enter image description here

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    $\begingroup$ Cheers for the memorabilia :) $\endgroup$ – probiner Jun 30 '17 at 14:31
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The last figure was also teached to me mnemotecnically to memorize the value of trigonometric functions on certain angles, notice that you can also calculate the tangent by ignoring dividing 2 this way:

Take the 0 and the 4 of the fist column (0 radians), divide and take the root, you have: $tan (0)= \sqrt(\dfrac{0}{4})=0$

For the second column we have $\dfrac{\pi}{6}$, take the 1 and the 3, divide and take the root again: $tan (\dfrac{\pi}{6})=\sqrt(\dfrac{1}{3})=\dfrac{\sqrt(1)}{\sqrt(3)}=\dfrac{1}{\sqrt 3}$

Similarily, $tan(\dfrac{\pi}{4})=\sqrt(\dfrac{2}{2})=\sqrt(1)=1$

$tan(\dfrac{\pi}{3})=\sqrt(\dfrac{3}{1})=\sqrt 3$

Finally, take for the 4th column ($\dfrac{\pi}{2}$) 4 and $0$. $tan(\dfrac{\pi}{2})=\sqrt(\dfrac{4}{0})!$ wich is in agreement of the fact that $tan$ is not defined on $\dfrac{\pi}{2}$ and thus cannot be.

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