Any Circle, Any Radius

On a circle of radius r the picture is the unit circle scaled by r. Every length grows by the same factor, so the point at angle θ is (r cos θ, r sin θ).

On a circle of radius 6 the point at 240° has x = $6\cos\left(240^{\circ}\right)$ and y = $6\sin\left(240^{\circ}\right)$. 240° is in quadrant III with reference angle 60°, so $\cos\left(240^{\circ}\right) = \frac{-1}{2}$ and $\sin\left(240^{\circ}\right) = \frac{-\sqrt{3}}{2}$. The point is ($-3$, $-3\sqrt{3}$).

A seat on a wheel or the tip of a fan blade is tracked the same way: the centre is the origin, the radius scales the unit circle, the angle picks the point.