Symmetry of a circle proof

Homework Statement

http://tng.trekcore.com/1.JPG [Broken]

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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Shooting Star
Homework Helper
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 [Broken]

Last edited by a moderator:
Shooting Star
Homework Helper
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.

HallsofIvy
Homework Helper

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 [Broken]

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Shooting Star
Homework Helper
(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.

Shooting Star
Homework Helper
EDIT: ignore

Shooting Star
Homework Helper
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.