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. :-)