Sparked by a conversation this past weekend about the usefulness of the half-angle identities, I constructed geometric proofs for and . 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.

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Though this one is my favorite:

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

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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!