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There exists a common trick to remember exact sine cosine and tangent values. The trick is relatively long, so instead of reposting it, please refer to my answer on this page.

Although I have used this trick for a while, I've never understood why it works. I understand the tangent values (sine/cosine according to right angle ratios) and the why the cosine values are in "the opposite order" of sine values (due to the cosine function being a $90^\circ$ phase shift on the sine function) but why does doing the above trick provide the correct values for sine (and cosine in opposite order)?

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This is just a briefer restatement of the same rule, which I find easier to remember. I don't think there is a particular reason why it works. (If there were, there would be some nice generalization, which there does not seem to be.)

$\theta \hskip2cm 0^\circ \hskip 1cm 30^\circ \hskip 1cm 45^\circ \hskip1cm 60^\circ \hskip1cm 90^\circ $

$\sin(\theta) \hskip1cm {\sqrt{0} \over 2}\hskip1cm {\sqrt{1}\over 2}\hskip1cm {\sqrt{2}\over 2}\hskip1cm {\sqrt{3}\over 2}\hskip1cm {\sqrt{4}\over 2}$

$\cos(\theta) \hskip1cm {\sqrt{4} \over 2}\hskip1cm {\sqrt{3}\over 2}\hskip1cm {\sqrt{2}\over 2}\hskip1cm {\sqrt{1}\over 2}\hskip1cm {\sqrt{0}\over 2}$

Edit: You might say that these values correspond to looking at triangles where the Pythagorean theorem becomes one of the following: $$4=0+4=1+3=2+2=3+1=4+0$$ and that these triangles happen to have very nice angles.

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  • $\begingroup$ But there has to be a reason! Mathematical phenomenon is based on relations not random "coincidences"! $\endgroup$ – user26649 Jul 15 '12 at 19:38
  • $\begingroup$ Of course there is a reason, it is just not a terribly interesting one: It all boils down to the use of Pythagoras on some particularly simple triangles. There is no mystery here, just a little bit of ingenuity in devising the pattern. $\endgroup$ – Harald Hanche-Olsen Jul 15 '12 at 19:51
  • $\begingroup$ @HaraldHanche-Olsen So basically it's just a observation that the values can be represented through the above trick? There's no reason, it's just an pattern devised for rememberance? Am I understanding it correctly? $\endgroup$ – user26649 Jul 15 '12 at 20:08
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    $\begingroup$ As aspiring writers are told, each plot is allowed one unexplained coincidence. The coincidence here is that the $\sqrt1$-$\sqrt3$-$\sqrt4$ triangle happens to have nice angles; that single triangle accounts for all of the 30° and 60° values. Since $4$ happens to be even we can of course represent a 45° triange with the same hypotenuse too, and the 0° and 90° values are just trivial. Note that the progression 0-30-45-60-90 is not particularly nice, though. $\endgroup$ – hmakholm left over Monica Jul 15 '12 at 22:07
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    $\begingroup$ @user26649 - Don't underestimate the power of mathematical serendipity. $\endgroup$ – steven gregory Aug 7 '18 at 15:44

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