What Are Polar Coordinates?

In polar coordinates, points are defined by a distance from the origin (\(r\)) and an angle (\(\theta\)), not by \(x\) and \(y\):

\[ x = r \cos \theta, \quad y = r \sin \theta \]

This system is ideal for curves that "spiral" or "loop" around a center.

Calculus in Polar Form

• Slope: Use calculus to find slopes and tangents in polar form.

• Area: The area inside a polar curve is

  \[ A = \frac{1}{2} \int_{\alpha}^{\beta} [r(\theta)]^2 d\theta \]

• Arc Length: Find curve lengths using integrals in polar form.

Where Are Polar Coordinates Used?

They're essential in modeling orbits, navigation, and even weather radar!