alt Hacker NewsThe problem is that trigonometric functions are used in many more fields beyond geometry. The input is not always an angle around a point in euclidean space, it could be phase angle of a periodic signal. You can make an alternative set of trig functions that take turns, but you will anger a lot of people if you mess with the vanilla trig functions.
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sriku • today at 2:56 AM
You'll have to bring in the 2π factor somewhere. Cant escape it. If sint is the sin function but with angle give in turns, then d/dx sint(x) = 2π cost(x). sin(x) ~ x for small x but sint(x) ~ 2πx for small x.
When dealing with waves you often are dealing with turns - or, as they’re called in that world, cycles. A cycle is a turn is tau is 2pi.
The SI unit for frequency after all is Hertz - cycles per second - which should really be considered equal to 2pi s^-1, but for complicated reasons, often isn’t, and most formulae that involve frequency ignore the ‘cycle’ - or it’s also hiding inside the definition of something like the wavelength or the Planck constant where it cancels out.
Meanwhile the SI unit for angular velocity is radians per second which is dimensionally equivalent to s^-1.
That said a becquerel, which measures rate of discrete events, is also dimensionally s^-1. (Next time you are measuring traffic to your website consider using the appropriate SI unit for measuring requests per second: the Becquerel.) - so dimensional equivalence isn’t the same as equivalence. You wouldn’t add a rate to a frequency, same as you probably shouldn’t add a torque to an amount of energy.