Originally Posted by

**Opalg** When I first read the original post in this thread, I saw the typo in the formula for $\displaystyle \cos(\omega_1+\omega_2)$ and I assumed that was the cause of the error. But clearly that was not the case, because it's only the sine formula that is used in the expression of y2.

It certainly is confusing, and that is because the question is confusingly worded. In fact, it asks about what happens when a *curve* is rotated (about the origin), but the rest of the post looks at what happens when a *point* is rotated.

If the *point* $\displaystyle (x_1,y_1)$ is rotated counterclockwise through an angle $\displaystyle \omega$ then it becomes the point $\displaystyle (x_2,y_2) = (x_1\cos\omega-y_1\sin\omega,x_1\sin\omega+y_1\cos\omega)$, which is essentially what you have calculated above.

But if a *curve* is rotated through an angle $\displaystyle \omega$ then its equation becomes transformed by replacing x by $\displaystyle x\cos\omega + y\sin\omega$, and y by $\displaystyle -x\sin\omega+y\cos\omega$, in the original equation of the curve. I guess that this is effectively because rotating the curve counterclockwise is equivalent to rotating the coordinate axes clockwise.