Polar Unit Vectors and the Velocity of Circular Motion

Draw the point $P$ at angle $\theta(t)$ on the circle, write $\vec{r}(t) = r\,\hat{r}(t)$, and record $\hat{r} = \cos\theta\,\hat{i} + \sin\theta\,\hat{j}$ and $\hat{\theta} = -\sin\theta\,\hat{i} + \cos\theta\,\hat{j}$.

Differentiate both decompositions with the chain rule and obtain $\frac{d\hat{r}}{dt} = \frac{d\theta}{dt}\,\hat{\theta}$ together with $\frac{d\hat{\theta}}{dt} = -\frac{d\theta}{dt}\,\hat{r}$.

Differentiate $\vec{r}(t) = r\,\hat{r}(t)$ at fixed radius, substitute $d\hat{r}/dt$, and define the tangential component $v_\theta = r\,d\theta/dt$ and the angular speed $\omega \equiv |d\theta/dt|$.