The typical way to define theta is to measure it down from the +z axis. There may be some special cases where you wouldn't do that, but down from the +z is standard.
Use spherical polar coordinates to show that the length of a path joining two points on a sphere of radius R is
I was initially told that phi should be rotation around the z-axis measured from the x-axis, and theta was the angle up out of the x-y plane. Working it this way I set up a triangle with hypotenuse dL and legs and . The following picture shows how I have the coordinates set up:
This gives me:
Which is what I was supposed to get except I have cosines instead of sines. This problem is easily solved by having be measured down from the z-axis instead of up from the x-y plane, but someone had told me to set up the coordinates as I specified above. I was hoping someone could either verify that what I've done is correct or point out where I messed up. Thanks.
I was looking for verification that the trig function that I got in my result is correct, that is, that I should have cos if measuring up and sin if measuring down. Although I think I got the answer, my book had a similar problem in which the angle was measured down from the z axis, which would give me sin, which is consistent with what they were asking me to prove. Thank you for the responses.