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.

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.

I could forget something but sin'(ax) = cos(ax). If a is a constant factor.

You need to apply the chain rules, ie

    sin'(ax) = a cos(ax)

Thank you. I'm really bad at calculus.

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.

Well this isn’t very fair. Yes, triangles have very little to do with the true nature of sin and cos.

It’s also true that they are the basic building blocks of cyclicity.

But to say they are not geometric tools is dishonest. They instead show us that geometry is deeply connected to many other, sometimes-surprising, areas of mathematics.

"triangles have very little to do with the true nature of sin and cos" <- this is what I mean.

Sure, but circles are geometric too :)

Absolutely, but IMO circles have as little to do with sin and cos as triangles do :-)

That seems ahistoric as per the meaning of the word. "Sine" is derived from the Sanskrit word for 'chord' as per its initial usage in determining the length of straight line segments between two arbitrary points on a circle.

cos and sin are the x and y coordinates as you turn around the unit circle in unit speed.

A turn doesn't have to represent an angle. It can also be a "cycle" in the oscillation.

I studied engineering, and pretty much everywhere where we needed the radian form with Pi, the mental reasoning was "one cycle, or repetition, or loop ot whatever is Pi". Never did Pi have any deeper meaning that helped understand the logic of the problem.

> 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)

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.

> the area is just a convenient geometric interpretation, which actually makes no sense for many (most?) integrals.

Could you elaborate on this? Because we defined the (Lebesgue) integral in my analysis 3 course exactly in this way: First define what measurable sets are, and what their volume is. Then the integral of a non-negative function is the volume under its graph, if that is measurable.

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.

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.