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Stacking triangles for fun and profit

One thing you may have noticed about the trigonometric functions sine and cosine is that they seem to have no agreed upon definition. Or rather, different authors choose different definitions as the starting point, mainly based on convenience. This isn’t problematic, or even particularly unusual in mathematics - as long as we can derive any of the other forms from any starting point, it makes little theoretical difference which we start from as they’re all equivalent anyway.

The most common starting points are the series definitions, the solution to an initial value problem involving ordinary differential equations, or using complex numbers as Euler’s formula. We don’t start from intuitive definitions and work on them until we can understand them more deeply or connect them to other structures; no, we simply write down a bizarre and unmotivated equation and show it has the desired properties, with no mention of how anyone thought it up in the first place. As the goal is to ground everything on the geometric definition in an easy-to-follow way, jumping ahead and using theorems before we’ve established that foundation defeats the purpose.

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