Coordinates of a point on a circle - review

Question

Given a rotation $θ$ and a radius $r,$ how do I find the coordinate $(x,y)$?

I saw an answer like this:

From the picture, it seems that your circle has centre the origin, and radius $r.$ The rotation appears to be clockwise. And the question appears to be about where the point $(0,r)$ at the top of the circle ends up.

The point $(0,r)$ ends up at $x=r\sin θ, y=r\cos θ.$

In general, suppose that you are rotating about the origin clockwise through an angle $θ.$ Then the point $(s,t)$ ends up at $(u,v)$ where $u=s\cos θ+t\sin θ$ and $v=−s\sin θ+t\cos θ.$

can someone please explain to me what happened? I'm confused :(

Thanks a lot!

Answer 1

We need simply to apply the definition of $\sin \theta$ and $\cos \theta$ and note that

• $x=r\cos \left(\theta-\frac{\pi}2\right)=r\cos \left(\frac{\pi}2-\theta\right)=r\sin\theta$

• $y=-r\sin\left(\theta-\frac{\pi}2\right)=r\sin \left(\frac{\pi}2-\theta\right)=r\cos\theta$

For the second part note that fro a clockwise rotation of $\theta$

• $(0,1)$ goes to $(\sin \theta, \cos \theta)$

• $(1,0)$ goes to $(\cos \theta,-\sin \theta)$

then a generic point

• $(s,t)=s(1,0)+t(0,1)$

goes to

• $(u,v)=s(\cos \theta,-\sin \theta)+t(\sin \theta, \cos \theta)=(s\cos \theta+t\sin \theta,-s\sin \theta+t\cos \theta)$