Rotation of cartesian coordinate system

[xzibition8612]

Homework Statement

Please see the rotation formula in the attachment.

Homework Equations

The Attempt at a Solution

I understand this formula rotates x,y into x',y' by some angle theta. Problem is, how is this formula derived? I cannot for the life of me visualize the cosine and sine transformation physically. Can someone explain to me how you get this formula. Thank you very much.

 

[gneill]

Consider the vector that extends from the origin to the point (x,y) in the base coordinate system. It has some magnitude R and angle β with respect to the x-axis of the coordinate system. In fact, x = Rcos(β) and y = Rsin(β).

Rotating that point around the origin by some angle θ is equivalent to rotating the vector by θ, so what would the coordinates of its endpoint be?

 

[xzibition8612]

so the end points would be x=Rcos(β+θ), y=Rsin(β+θ). Then what? I still can't see how this relates to the formula, espcially how in the formula for x' and y' individually there are x and y terms together.

 

[Muphrid]

What you just found are the x' and y' coordinates. Expand the sines and cosines using angle sum formulas and put any sines or cosines of \beta in terms of the origina x,y.

 

[xzibition8612]

x' = R(cosβcosθ-sinβsinθ)
y' = R(sinβcosθ+sinβcosθ)

x' = R[(x/R)cosθ-(y/R)sinθ]
y' = R[(y/R)cosθ+(y/R)cosθ]

arrrrgh almost there. First term in y' is wrong. I get y' = ycosθ ... instead of y' = xsinθ ...Can someone point out myt mistake? Thanks a lot for your help!

 

[Muphrid]

In your second line, you forgot to switch beta and theta in the second term.