Proving Circle Symmetry through Rotation: Understanding the Equations

AI Thread Summary
The discussion focuses on proving the symmetry of a circle using the equation x² + y² = a² through rotation. Participants are trying to derive the transformation equations x = x'cosθ - y'sinθ and y = x'sinθ + y'cosθ, which relate coordinates before and after rotation. The method involves dropping perpendiculars from points on the circle to the axes and applying properties of similar and right triangles. A key point of confusion is how to derive the -y'sinθ term, with users seeking clarification on their geometric approach. The conversation highlights the importance of visual aids and proper diagramming in understanding the proof.
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!
 
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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
 
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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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