Regardless of relative directions around circles, the sine and cosine between 0° and 90° can be described unambiguously in terms of ratios between side lengths of right triangles. You could decide to define the functions differently, but then they'd no longer be the sine and cosine, they'd be something else. The whole point of having these two named functions is that they're a common ground, independent of whatever idiosyncratic angle measurements are useful for a given problem.

> Regardless of relative directions around circles, the sine and cosine between 0° and 90° can be described unambiguously in terms of ratios between side lengths of right triangles.

That's true, if there are no angles greater than 90° or less than 0°, as is the case in a non-pathological right triangle. In this case, as ratios of nonnegative lengths, all trig functions are always nonnegative.

If you want to include angles outside those bounds, then you care about what exactly occurs where, and while you can unambiguously define angles between 0 and 90 to have all positive trig functions, you can also unambiguously define them to have negative sines and tangents. Fundamentally what's happening is that you're defining certain line segments to have negative length instead of positive length. Which line segments should have negative length isn't a question about angles.

> You can decide to define the functions differently, but then they'd no longer be the sine and cosine, they'd be something else.

Only in a sense much stricter than what people generally use. Sine and cosine themselves are hard to distinguish - you can also call them sine (x) and sine (x - 270). Some people might argue that the sine of (x - 270) is still a sine.

> In general, the two functions can be described by their differential equations

If you do that, you'll completely lose the information about where sine is positive and where it's negative. You can apply any phase shift you want (as long as you apply it to both functions) and their differential equations will look exactly the same.

> If you want to include angles outside those bounds, then you care about what exactly occurs where, and while you can unambiguously define angles between 0 and 90 to have all positive trig functions, you can also unambiguously define them to have negative sines and tangents.

You could define trig functions differently, but then you'd need a separate pair of unnamed functions to express "the ratios of unsigned side lengths of a right triangle in terms of its unsigned interior angles". It's the same reason we don't count "-1 apple, -2 apples, -3 apples, ...". Or why horizonal and vertical lines usually fall on the x-axis and y-axis instead of the (1/√2,1/√2)-axis and (-1/√2,1/√2)-axis. We optimize for the common case.

> If you do that, you'll completely lose the information about where sine is positive and where it's negative. You can apply any phase shift you want (as long as you apply it to both functions) and their differential equations will look exactly the same.

What do you mean? "sin(0) = 0, cos(0) = 1, and for all x, sin'(x) = cos(x), cos'(x) = -sin(x)" is perfectly unambiguous. If you changed the initial conditions, you'd get another pair of functions, but then they'd no longer be the sine and cosine, they'd be some other linear combination. And for that, refer to what I said about the x-axis and y-axis: better to take the stupid simple (0,1) solution and build more complex ones from there.

> What do you mean? "sin(0) = 0, cos(0) = 1, and for all x, sin'(x) = cos(x), cos'(x) = -sin(x)" is perfectly unambiguous.

It's pretty straightforward. "sin(0) = 0" is not a differential equation. Any phase shift applied to sine and cosine will produce exactly the same set of differential equations that apply to sine and cosine; you can rename the shifted functions "sine" and "cosine" and you'll be fine.

> That is an established mathematical convention, called "bearing".

Bearing is a nautical convention not a mathematical one.

I have worked on boat computer systems and can assure you that all the angles were in radians going in the proper direction while beatings were separate always shown in degrees and clockwise.

As someone else noted, "bearing" isn't a mathematical convention. To be clear, when I say "mathematical convention" I mean what mathematicians do and teach. But I didn't know about "bearing" in navigation so thanks for mentioning that.

> There isn't really one mathematical convention on "angles".

There is for angles in the plane, which are the angles I was discussing. In every math course from trig where people first encounter angles in the plane they increase as you go counterclockwise. This is true in trig, precalc, calculus, ... You will not find a math textbook in which plane angles increase clockwise. I think that counts as a convention.

That convention determines the graph of the sine function, because sin theta is defined in trig courses as the y-coordinate of the point where the ray from the origin determining the angle intersects the unit circle. So (e.g.) if 45 degrees means 45 degrees clockwise, that ray is below the x-axis, and the y-coordinate of the intersection is negative -- and hence, sin 45 degrees would be negative.

If angles increase clockwise from the positive x-axis, then sin 45 degrees will be negative. And if sine 45 degrees is negative, then angles are increasing clockwise from the positive x-axis. And any mathematician would tell you that sine 45 degree is 1/sqrt(2), not -1/sqrt(2).

> ... in a math class it's normal to orient phi in whatever way makes sense to you.

You're correct that there are two prevailing conventions for the angle phi in spherical coordinates. Mathematicians measure phi downward from the positive z-axis, so it takes values from 0 to 180 degrees. (Actually, it's sort of like "bearing" that you mentioned.) Physicists measure phi upward from the x-y plane, so it can take values from -90 to 90 degrees. It does cause some confusion in teaching Calc 3, because students also taking a physics or astronomy course may be seeing two conventions for phi. However, in 3 dimensions (spherical coordinates) there's no natural "clockwise" or "counterclockwise".

But there is a convention for measuring phi in math classes -- it's the one I described above. Check any calculus book. Our colleagues in physics don't like it, but oh well. :-)