> 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

> 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.