Expansion of a rotation-formula

**osudude** · January 14th 2010, 12:19 PM

Here is the question:

By expanding out $(rcos(\alpha + \theta), rsin(\alpha + \theta))$ , verify the formula:

$p_{\theta}(x,y) = (xcos(\theta) - ysin(\theta), xsin(\theta) + ycos(\theta))$

Ok, now I believe that $x=rcos(\alpha)$ and $y=rsin(\alpha)$ . Also, if you expand out the first formula you get

$(rcos(\alpha) + rcos(\theta), rsin(\alpha) + rsin(\theta)) = (x + rcos(\theta), y + rsin(\theta))$

but from here I am stuck. Am I onto something here or am I going at this all wrong? thanks.

**HallsofIvy** · January 14th 2010, 12:45 PM

Originally Posted by osudude

Here is the question:

By expanding out $(rcos(\alpha + \theta), rsin(\alpha + \theta))$ , verify the formula:

$p_{\theta}(x,y) = (xcos(\theta) - ysin(\theta), xsin(\theta) + ycos(\theta))$

Ok, now I believe that $x=rcos(\alpha)$ and $y=rsin(\alpha)$ . Also, if you expand out the first formula you get

$(rcos(\alpha) + rcos(\theta), rsin(\alpha) + rsin(\theta)) = (x + rcos(\theta), y + rsin(\theta))$

No, you don't. $cos(\alpha+ \theta)$ is NOT equal to $cos(\alpha)+ cos(\theta)$ , it is equal to $cos(\alpha)cos(\theta)- sin(\alpha)sin(\theta)$ .

but from here I am stuck. Am I onto something here or am I going at this all wrong? thanks.

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