Definition of Polar Coordinates

Theory

Polar coordinates represent a point in the plane using a distance from the origin and an angle from the positive x-axis.

They consist of two components:

$r$ : the radial distance from the origin to the point.
$\theta$ : the angle measured in radians from the positive x-axis to the line segment connecting the origin to the point.

The relationship between Cartesian coordinates $(x, y)$ and polar coordinates $(r, \theta)$ is given by:

Exercise 1

Convert the Cartesian coordinates $(3, 4)$ to polar coordinates.

Show solution ↓

Solution

To convert the Cartesian coordinates $(3, 4)$ to polar coordinates, we calculate the radial distance $r$ and the angle $\theta$ .

r = \sqrt{3^2 + 4^2} = 5, \quad \theta = \tan^{-1}\left(\frac{4}{3}\right) \approx 0.9273 \text{ radians}