Geometric Proofs of Trigonometric Identities

Sparked by a conversation this past weekend about the usefulness of the half-angle identities, I constructed geometric proofs for \sin(x/2) and \cos(x/2). Since I’ve never seen these anywhere before, I thought I’d share.

And while I was at it, I thought I’d share all my other geometric proofs, so here they are, posted mostly without comment.

Some of these are so well-known as to be not worth mentioning. Many of them have been stolen from Proofs Without Words I or Proofs Without Words II. I came up with a few of them myself. Frustratingly, almost none of them are to be found in Precalculus textbooks, where they might be learned and appreciated.

Pythag 1

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Pythag 2

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Pythag 3

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Pythag 4

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Sincos of a sum 1

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sincos of a diff 1

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sincos of a diff 2

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Though this one is my favorite:

sine and cosine of a sum best 1

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sine and cosine of a sum best 2

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Partially because of the way it naturally generalizes into the proof of the derivative of sine. If you just let \beta approach 0, \cos(\beta) approaches 1 and that point in the interior of the circle ends up on the circle, where \sin(\beta) merges with \beta itself.

Proof of derivative of sinx

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double angle 1

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double angle 2

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double angle 3

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half angle 1

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half angle 2

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half angle 3

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half angle 4____________________________________________________________________________________

And finally, one that shows that the sum of a sine and cosine function of the same argument is also a sinusoid. Since I lost the original picture and don’t feel like remaking it, you’ll have to complete the proof on your own!

sum of sine and cosine

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Update: After some feedback on twitter, I’ve decided to add a few more diagrams. Tim Brzezinski sent me a link to his website of geometric proofs of trig identities and he had some that I’ve never seen before.

Check it out!

https://www.geogebra.org/m/DxAcj8E2#material/QedMT7Pw

I’ve taken two of his diagrams and added them below.

tan of a sum 1

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tan of a sum 2

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tan of a diff 1

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tan of a diff 2

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