# Thread: Expansion of a rotation-formula

1. ## Expansion of a rotation-formula

Here is the question:

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

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

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

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

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

2. Originally Posted by osudude
Here is the question:

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

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

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

$\displaystyle (rcos(\alpha) + rcos(\theta), rsin(\alpha) + rsin(\theta)) = (x + rcos(\theta), y + rsin(\theta))$
No, you don't. $\displaystyle cos(\alpha+ \theta)$ is NOT equal to $\displaystyle cos(\alpha)+ cos(\theta)$, it is equal to $\displaystyle cos(\alpha)cos(\theta)- sin(\alpha)sin(\theta)$.

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