Polar coordinates and Cartesian coordinates are two different ways to show points on a flat surface. Each method has its own benefits.

Polar Coordinates:

• They are written as $(r, \theta)$, where:
  • $r$ is how far the point is from the center (the origin).
  • $\theta$ is the angle measured from the right side (the positive x-axis).

Example:

Let’s change the polar point $(5, \frac{\pi}{3})$ into Cartesian coordinates:

• We can use these formulas:
  • $x = r \cos(\theta)$
  • $y = r \sin(\theta)$

Calculating:

• For $x$:
  $x = 5 \cos\left(\frac{\pi}{3}\right) = 5 \cdot \frac{1}{2} = 2.5$

• For $y$:
  $y = 5 \sin\left(\frac{\pi}{3}\right) = 5 \cdot \frac{\sqrt{3}}{2} \approx 4.33$

So, the polar point $(5, \frac{\pi}{3})$ is about $(2.5, 4.33)$ in Cartesian coordinates.

Cartesian Coordinates:

• They are shown as $(x, y)$, where:
  • $x$ tells us how far to go left or right.
  • $y$ tells us how far to go up or down.

Example:

Let’s convert the Cartesian point $(3, 3\sqrt{3})$ into polar coordinates:

• We can use these formulas:
  • $r = \sqrt{x^2 + y^2}$
  • $\theta = \tan^{-1}\left(\frac{y}{x}\right)$

Calculating:

• For $r$:
  $r = \sqrt{3^2 + (3\sqrt{3})^2} = \sqrt{9 + 27} = \sqrt{36} = 6$

• For $\theta$:
  $\theta = \tan^{-1}\left(\frac{3\sqrt{3}}{3}\right) = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3}$

Thus, the Cartesian point $(3, 3\sqrt{3})$ can be written as $(6, \frac{\pi}{3})$ in polar coordinates.

This comparison shows how the two systems are connected. This can help make calculations or visualizations easier depending on what we need.