Angle of Tangent to Radius in Polar Coordinates

Theorem

Let $C$ be a curve embedded in a plane defined by polar coordinates.

Let $P$ be the point at $\left\langle{r, \theta}\right\rangle$.

Then the angle $\psi$ made by the tangent to $C$ at $P$ with the radial coordinate is given by:

$\tan \psi = r \dfrac {\mathrm d \theta} {\mathrm d r}$

Proof

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 1.3$: Families of Curves. Orthogonal Trajectories