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"math never decreed that sine and cosine have to take radian arguments!" This is at best questionable and at worst false. If you only want to use sin and cos

"math never decreed that sine and cosine have to take radian arguments!" This is at best questionable and at worst false. If you only want to use sin and cos as functions for doing trigonometry, it is true that you can choose whatever angle unit you like and stick with it and it will be fine. For most other stuff, e.g. differential equations, complex analysis, signal processing and mechanics, it's pretty much inescapable that the zeroes of sin are at integer multiples of pi, and that's that.

Double_a_92 4y ago
Why do they need to be at multiple integers of pi, instead of at half turns?
- qsort 4y ago
  Because - numeric precision arguments aside, there's an excellent comment explaining that problem in this thread - it's the only unit of measurement that makes sense for angles. It provides an easy way to connect the complex exponential with trigonometric functions (and everything you get from that, i.e. Taylor series, nice behavior in diffeqs). You can do the same in degrees as well, but you end up with weird conversion factors with pi in the denominator, a strong indication you should have multiplied by pi to begin with.
- cillian64 4y ago
  If you differentiate sin(x) with respect to x then you get cos(x), but only if your trig functions are using radians. Any other unit results in an extra coefficient appearing. That’s not an insurmountable problem, but radians are the fundamental unit here, not just an arbitrary choice.
  - bvrmn 4y ago
    I could forget something but sin'(ax) = cos(ax). If a is a constant factor.
    - cygx 4y ago
      You need to apply the chain rules, ie sin'(ax) = a cos(ax)
      - bvrmn 4y ago
        Thank you. I'm really bad at calculus.
  - 6gvONxR4sf7o 4y ago
    You get extra coefficients appearing, but also extra coefficients disappearing. If 2 pi x is showing up everywhere, you still have to deal with it: d/dx sin(2 pi x) = 2 pi cos(2 pi x) vs d/dy new_sin(y) = 2 pi new_cos(y) “Fundamental unit” really depends on what you care about.
- MayeulC 4y ago
  It's effectively the same, but cos, sin and others are defined with radians as arguments. Taylor's series expansions are changed, otherwise. So is Euler's formula.
- mjw_byrne 4y ago
  The main thing to realise is that sin and cos are not fundamentally tools for doing geometry. The fact that you can use them for working out side lengths of triangles or converting polar to cartesian coordinates is somewhat incidental. It doesn't help that at school our first look at sin and cos is all about adjacent sides and opposite sides in right-angled triangles. It's understandable, because jumping straight into the deep end would be too hard, but it's a bit misleading. In most mathematical applications, the x in "sin(x)" doesn't even represent an angle, so it doesn't make sense to talk about whether sin and cos are "in degrees or in radians". They're simply functions that crop up as solutions to the differential equation that describes harmonic oscillation; or the imaginary and real parts of e^ix; or exponentiation of certain matrices; or a whole load of other stuff I haven't thought of. In all those settings, it turns out that sin has zeroes at integer multiples of pi, which forces the convention that a half-turn is an angle of pi, and the definition of radians follows from there. But as I said, for the specific case of basic trig, carrying around a scaling factor and doing everything in degrees is easy enough. Carrying that same scaling factor around in pretty much any other application of sin, cos and related functions would be hell.
  - red_trumpet 4y ago
    > or the imaginary and real parts of e^ix For that particular application, x is exactly the argument[1] of your complex number, though! [1] https://en.wikipedia.org/wiki/Argument_(complex_analysis) https://en.wikipedia.org/wiki/Argument_(complex_analysis)
    - mjw_byrne 4y ago
      True, however that doesn't mean x represents an angle. It means you can put a geometric interpretation on an abstract formula. e^ix = cos x + i sin x regardless of what x represents - if you were doing electrical engineering it might be time, for example. Interpretation is often strongly motivated by what "comes first" in the order in which you learn things, so the whole debate is a bit subjective anyway. Another common example: at school we learn the integral is the area under the curve. At university we learn the area under the curve is the integral. The integral is the "real" thing and the area is just a convenient geometric interpretation, which actually makes no sense for many (most?) integrals. At school we learn it "backwards" purely because it's easier that way, and visual aids are helpful. I think something similar applies to sin and cos.

Double_a_92 4y ago

Why do they need to be at multiple integers of pi, instead of at half turns?

qsort 4y ago
Because - numeric precision arguments aside, there's an excellent comment explaining that problem in this thread - it's the only unit of measurement that makes sense for angles. It provides an easy way to connect the complex exponential with trigonometric functions (and everything you get from that, i.e. Taylor series, nice behavior in diffeqs). You can do the same in degrees as well, but you end up with weird conversion factors with pi in the denominator, a strong indication you should have multiplied by pi to begin with.
cillian64 4y ago
If you differentiate sin(x) with respect to x then you get cos(x), but only if your trig functions are using radians. Any other unit results in an extra coefficient appearing. That’s not an insurmountable problem, but radians are the fundamental unit here, not just an arbitrary choice.
- bvrmn 4y ago
  I could forget something but sin'(ax) = cos(ax). If a is a constant factor.
  - cygx 4y ago
    You need to apply the chain rules, ie sin'(ax) = a cos(ax)
    - bvrmn 4y ago
      Thank you. I'm really bad at calculus.
- 6gvONxR4sf7o 4y ago
  You get extra coefficients appearing, but also extra coefficients disappearing. If 2 pi x is showing up everywhere, you still have to deal with it: d/dx sin(2 pi x) = 2 pi cos(2 pi x) vs d/dy new_sin(y) = 2 pi new_cos(y) “Fundamental unit” really depends on what you care about.
MayeulC 4y ago
It's effectively the same, but cos, sin and others are defined with radians as arguments. Taylor's series expansions are changed, otherwise. So is Euler's formula.
mjw_byrne 4y ago
The main thing to realise is that sin and cos are not fundamentally tools for doing geometry. The fact that you can use them for working out side lengths of triangles or converting polar to cartesian coordinates is somewhat incidental. It doesn't help that at school our first look at sin and cos is all about adjacent sides and opposite sides in right-angled triangles. It's understandable, because jumping straight into the deep end would be too hard, but it's a bit misleading. In most mathematical applications, the x in "sin(x)" doesn't even represent an angle, so it doesn't make sense to talk about whether sin and cos are "in degrees or in radians". They're simply functions that crop up as solutions to the differential equation that describes harmonic oscillation; or the imaginary and real parts of e^ix; or exponentiation of certain matrices; or a whole load of other stuff I haven't thought of. In all those settings, it turns out that sin has zeroes at integer multiples of pi, which forces the convention that a half-turn is an angle of pi, and the definition of radians follows from there. But as I said, for the specific case of basic trig, carrying around a scaling factor and doing everything in degrees is easy enough. Carrying that same scaling factor around in pretty much any other application of sin, cos and related functions would be hell.
- red_trumpet 4y ago
  > or the imaginary and real parts of e^ix For that particular application, x is exactly the argument[1] of your complex number, though! [1] https://en.wikipedia.org/wiki/Argument_(complex_analysis) https://en.wikipedia.org/wiki/Argument_(complex_analysis)
  - mjw_byrne 4y ago
    True, however that doesn't mean x represents an angle. It means you can put a geometric interpretation on an abstract formula. e^ix = cos x + i sin x regardless of what x represents - if you were doing electrical engineering it might be time, for example. Interpretation is often strongly motivated by what "comes first" in the order in which you learn things, so the whole debate is a bit subjective anyway. Another common example: at school we learn the integral is the area under the curve. At university we learn the area under the curve is the integral. The integral is the "real" thing and the area is just a convenient geometric interpretation, which actually makes no sense for many (most?) integrals. At school we learn it "backwards" purely because it's easier that way, and visual aids are helpful. I think something similar applies to sin and cos.