Where do the rotation coordinate transformation equations come from?

[raintrek]

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!

 

[Shooting Star]

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.

 

[raintrek]

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

 

[Shooting Star]

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]

Just out of curiosity, why was this posted under "Introductory Physics"?

 

[raintrek]

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

 

[Shooting Star]

(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]

EDIT: ignore

 

[Shooting Star]

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.