To change Cartesian coordinates to polar coordinates, you need to know how these two systems relate to each other.

What Are Coordinates?
In Cartesian coordinates, you describe a point in a 2D space using $(x, y)$.
In polar coordinates, the same point is shown as $(r, \theta)$.
Here, $r$ is the distance from the center (origin), and $\theta$ is the angle you measure from the positive x-axis.

Here are the main steps to convert them:

1. Finding the Radius ($r$):
   To find $r$, use this formula:

   $$r = \sqrt{x^2 + y^2}$$

   This formula comes from the Pythagorean theorem. It shows the longest side (hypotenuse) of a right triangle made with the coordinates.

2. Finding the Angle ($\theta$):
   You can find the angle $\theta$ using the tangent function:

   $$\theta = \tan^{-1}\left(\frac{y}{x}\right)$$

   It's important to check the signs of $x$ and $y$ to figure out which quadrant (section) $\theta$ is in. This is because the inverse tangent function can give answers in both the first and fourth quadrants.

3. Adjusting for Quadrants:
   Based on the signs of $x$ and $y$, you might need to adjust $\theta$:

   • If $x > 0$ and $y \geq 0$: $\theta = \tan^{-1}\left(\frac{y}{x}\right)$
   • If $x < 0$: $\theta = \tan^{-1}\left(\frac{y}{x}\right) + \pi$
   • If $x > 0$ and $y < 0$: $\theta = \tan^{-1}\left(\frac{y}{x}\right) + 2\pi$ (to keep it a positive angle)

4. Final Polar Coordinates:
   After you find $r$ and $\theta$, you can write the polar coordinates as $(r, \theta)$.

In short, changing from Cartesian to polar coordinates means calculating how far the point is from the center (the radius) and figuring out the angle based on the y and x coordinates. This process is important for understanding shapes in polar coordinates, especially in topics like calculus when working with integration in polar form.