alt Hacker NewsI like to store angles as turns in my own code, because (as noted) it makes quarter-turns computable without rounding. OTOH if you need, say, twelfths of a turn, you might want to just store angles as degrees since that’s already common.
Michael Spivak, in Calculus (3rd ed p. 301) considers the unit choice to be a property of the function and initially defines sin° and sinʳ (before settling on sin meaning sinʳ) and considers “sin x°” and “sin x radians” to be misleading, saying that ‘a number x is simply a number—it does not carry a banner indicating that it is “in degrees” or “in radians”’. I don’t really understand this argument, since in science and engineering we constantly carry units around with our quantities.
Replies
It's because both radians and degrees are are a ratio of a length to another length and are thus dimensionless. No matter how you measure it, all angles are without a unit.
It's most obvious with radians but it's also the case with degrees.
Using radians, you are guaranteed to not introduce unusual extra terms to rescale angles, if you use any other scale of angle you will have to keep track of extra terms.
That may be useful in whatever you're doing. I work in degrees quite often and I'm careful to keep track of the 2pi/360 terms that crop up all over the place. With grade measure you have to keep track of 2pi/400 terms and with turns you have to keep track of 2pi/1 terms that will repeatedly show up.
Again, depending on what you're doing, this may or may not make sense to do.
In general, mathematics works out easier when the scaling term is 2pi/2pi because then you have a lovely 1 scale factor you don't have to keep track of.
You generally can’t apply functions to dimensional units. The only thing units can do is be multiplied or divided together. So I can multiply a mass by a distance or divide a distance by a speed, and I can multiply the result by a scalar; but I can’t take the sine of a distance or the logarithm of a time or exponentiate a mass. Those are things I can only do to scalars.
‘But wait!’ You may cry: ‘the formula for a transverse wave varies with the sine of a distance!’
To which I would say no: it varies with the sine of a distance (the horizontal displacement), divided by another distance (the wavelength), divided by 2pi. The distances cancel out and leave a scalar. The sine is taken of that pure scalar; it results in a pure scalar; and then it’s multiplied by another distance (the amplitude) to give you a vertical displacement. Sine is a pure function.
Something else to consider is that the way we combine units with scalars to create dimensional quantities is through multiplication - and it’s not like there’s a simple formula for what a sine of a product is - I can’t determine sin(ab) in terms of sines or other functions of a and b. So if, say, a ‘degree’ were some dimensional unit, sin(90°) would not be something I could calculate - despite knowing sin(90) I don’t know sin(°) - whatever that would mean - and even if I did it gets me no closer to figuring out sin(90°)
Realizing that ° is just a mathematical constant equal to pi/180 solves a lot here.