This is one of those ideas that almost certainly exists already, but I’m so proud that I just have to tell you about it, internet. It’s remarkably simple, as my revelations tend to be.
Because any trigonometric function can be represented as a ratio of two sides of a right triangle (for example, the sine of an angle is the ratio of the opposite leg per the hypotenuse), it ought to be possible to prove any trigonometric identity with a clever picture. No algebra needed. You could use algebra to describe the proof, but it wouldn’t be the proof itself. You just “see” the truth of the identity in an instant.
It’s like the proof in Lockhart’s lament,
For example, if I’m in the mood to think about shapes— and I often am— I might imagine a
triangle inside a rectangular box:
I wonder how much of the box the triangle takes up? Two-thirds maybe?
If I chop the rectangle into two pieces like this, I can see that each piece is cut diagonally in half by the sides of the triangle. So there is just as much space inside the triangle as outside. That means that the triangle must take up exactly half the box!
This is what a piece of mathematics looks and feels like. That little narrative is an example of the mathematician’s art: asking simple and elegant questions about our imaginary creations, and crafting satisfying and beautiful explanations. There is really nothing else quite like this realm of pure idea; it’s fascinating, it’s fun, and it’s free!