> 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'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.