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eyesontheball1
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Homework Statement
I'm feeling a bit ambivalent about my interpretation of spherical coordinates and I'd appreciate it if someone could clarify things for me! In particular, I'd like to know whether or not my derivation of the coordinates is legitimate.
Homework Equations
Considering only the xy-plane, x = rcos(θ), y = rsin(θ) s.t. r ≥0, -π≤θ≤π.
Now, if we introduce the z-axis, so that we're in 3-dimensional space, we can construct a right triangle s.t. the base of the triangle lies in the xy-plane and the right angle of the triangle is formed between the base of the triangle and the side of the triangle pointing upward into the positive z direction normal to the xy-plane. The hypotenuse of the triangle lying in the xy-plane has length r. This hypotenuse is also the base of the second triangle. We choose ø s.t. -π/2 ≤ ø ≤ π/2, and we let r = pcos(ø), so that the length of the base of the second triangle is r = pcos(ø), and we also let the length of the side of the triangle normal to the xy-plane be z = psin(ø). It then follows that the length of the hypotenuse of the second triangle is p. What I'm unsure about is whether or not it's okay to interpret the angle, ø, as being the angle formed between the xy-plane and the hypotenuse of the second triangle, given a fixed point, (x,y), in the xy-plane. Thanks in advance!