# Angle of Tangent to Radius in Polar Coordinates

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## Theorem

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

Let $P$ be the point at $\polar {r, \theta}$.

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

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

## Proof

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## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 3$: Families of Curves. Orthogonal Trajectories