Proving Circle Symmetry through Rotation: Understanding the Equations

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Homework Help Overview

The original poster attempts to prove the symmetry of a circle by demonstrating that the equation x² + y² = a² holds true under rotation. They reference specific equations related to the rotation of axes but express uncertainty about their derivation based on a provided diagram.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants discuss using geometric properties, such as similar triangles and right-angled triangles, to derive the equations for rotated coordinates. There are questions about the specific terms in the equations and how they relate to the geometry of the situation.

Discussion Status

Some participants are actively engaging with the problem, offering suggestions for geometric constructions and questioning the original poster's understanding of the equations. There is a sense of progress as the original poster indicates they are close to resolving their confusion, although no consensus or final solution has been reached.

Contextual Notes

There is mention of diagrams that are not visible to some participants, which may affect the clarity of the discussion. Additionally, there is a question about the appropriateness of the thread's categorization under "Introductory Physics."

raintrek
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Homework Statement


http://tng.trekcore.com/1.JPG

I'm trying to prove that the circle is symmetrical by showing that x² + y² = a² holds when the circle rotates.

I know that this is proved given the following two formulae:
x = x'cosθ - y'sinθ
y = x'sinθ + y'cosθ

but I don't know where those two equations have come from based on my diagram. Help!
 
Last edited by a moderator:
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Drop a perpendicular from where the x' axis cuts the circle to the x-axis and another perp from where the y' axis cuts the circle to the y-axis. Use some properties of similar triangles and right angled triangles.
 
I've got the x'cosθ part of the expression for x, but I just cannot see how the -y'sinθ is found...

http://tng.trekcore.com/1.GIF
 
Last edited by a moderator:
My mistake for giving a hasty answer. Sorry.

Take a point P:(x,y) in the x-y system. Now draw x' and y' axes, rotated by some theta. If you drop perps from P on the x-axis and the x' axis, the first perp cuts the x-axis at a dist x from O and the 2nd perp cuts the x'-axis at a dist x' from O. Now, find x in terms of x' and y', using elementary geometry.
 
Just out of curiosity, why was this posted under "Introductory Physics"?
 
OK, i think I'm almost there,

I have the x'cosθ term, and I know I need to minus the purple section, which I trust is y'sinθ -- but I can't seem to show that it is, lol, it's the last stumbling block

http://tng.trekcore.com/2.GIF
 
Last edited by a moderator:
(HallsofIvy has asked you a question. I am also curious.)

Have you drawn the diagram as I said in my 2nd post? You can show us, if possible.
 
EDIT: ignore
 
Hi raintrek,

I'm not able to see the pictures you posted initially. Have you removed them, or is something wrong with my browser settings? Please answer asap.
 

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