Spherical Coordinates: Understanding Theta Equation

AI Thread Summary
The discussion centers on the correct formulation of the theta equation in spherical coordinates. The original equation presented on the referenced page is incorrect; the correct equation is θ = tan^(-1)(√(x² + y²)/z). An alternative and more commonly used formulation is θ = cos^(-1)(z/r), where r = √(x² + y² + z²). The preference for the cos^(-1) formulation may relate to its broader applicability in higher mathematics. Overall, the conversation highlights the importance of accurate representations in mathematical equations.
MattRob
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So, I was curious about this and found more or less what I was looking for here: http://electron9.phys.utk.edu/vectors/3dcoordinates.htm

Except, something is bothering me about those equations. At the very bottom, the equation for Theta in a spherical coordinate system; shouldn't it be
\theta = {tan^{-1}}( \frac{\sqrt{{x^{2}}+{y^{2}}}}{z})
instead of
\theta = {tan^{-1}}( \frac{z}{\sqrt{{x^{2}}+{y^{2}}}})

(The image in question)
p22.gif


Because {tan^{-1}}( \frac{opposite}{adjacent}) = \theta , and looking at angle \theta , the line opposite of it is exactly equal to \sqrt{{x^{2}}+{y^{2}}} , and the line adjacent to it equal to z.

So I'm wondering if I'm in error (and how so if I am) or if the linked page is.
 
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You are correct. That equation on that page is in error.
 
It appears that page is wrong. You're right, the correct relation is

##\theta = \tan ^{-1}\left( \frac{\sqrt{x^2 +y^2 }}{z}\right)##.

You could also write it as ##\theta = \cos ^{-1} \left(\frac{z}{\sqrt{x^2 +y^2 +z^2 }}\right)##Edit: Darn you, jtBell, you beat me by seconds!
 
yep... it sure looks like they've got their sides mixed up.
However, it is more usual to use ##\theta = \cos^{-1}(z/r):r=\sqrt{x^2+y^2+z^2}##
 
Thanks very much for all the replies! Are the equations for the x, y, and z components from a spherical coordinate system correct on that page, though?

And why is the ##\theta = \cos ^{-1} \left(\frac{z}{\sqrt{x^2 +y^2 +z^2 }}\right)## approach more common? The \theta = {tan^{-1}}( \frac{\sqrt{{x^{2}}+{y^{2}}}}{z}) one has less terms. Something to do with higher mathematics?
 
So I know that electrons are fundamental, there's no 'material' that makes them up, it's like talking about a colour itself rather than a car or a flower. Now protons and neutrons and quarks and whatever other stuff is there fundamentally, I want someone to kind of teach me these, I have a lot of questions that books might not give the answer in the way I understand. Thanks
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