[justinpombrio]

This is a very good point, but it took me a minute to get what you were saying beneath the snark. Translating without the snark:

There's a famous equation relating sin and cos to complex exponentiation. It also helps explain the Taylor expansions of sin and cos, which is one way to compute them and to find properties about them. It's a very important equation. It is:

     ix
    e    = cos x   +   i sin x

kazinator's point was that this equation relies on cos and sin taking radians as arguments. If they take turns instead, then you need to insert messy extra constants to state this equation!

jVinc's counter-point, made with lots of snark, is that there's an equation that's even nicer if you just instead measure angles in turns with ncos and nsin:

    (-1)^(2x) = ncos(x) + i nsin(x)

It's similar, but doesn't require the magic constant e.

A proof sketch that these are equivalent:

    (-1)^(2x) = e^ln((-1)^(2x)) = -e^(2x) = e^(i * (2 pi x)) using e^(pi i) = -1

[thrtythreeforty]

Wow, this version makes intuitive sense when retroactively applying what I already know about the complex plane, and rotations about it. Thank you!

[areyousure]

Perhaps an even nicer equation:

    1^x = ncos(x) + i nsin(x)

using a multi-valued definition of the exponentiation on the left hand side.

[2muchcoffeeman]

Isn’t this just changing units to suit your purpose?

Same way we might use electron volts rather than volts to make the equations nice.

[justinpombrio]

Yes, it's exactly that. Changing the units from radians to turns, to make your equations nicer, because turns are evidently the more natural unit.