To change Cartesian coordinates to polar coordinates, we first need to understand how these two systems work.

Cartesian Coordinates

These are written as $(x, y)$ and show where a point is based on how far it is from a starting point called the origin. We measure distances horizontally (left or right) and vertically (up or down).

Polar Coordinates

On the other hand, polar coordinates are written as $(r, \theta)$. They describe a point using:

• r: The distance from the origin.
• θ: An angle that shows where the point is in relation to a starting direction, usually the positive x-axis (to the right).

Definitions

1. Polar Radius (r): This tells us how far the point is from the origin. We can figure it out using the Pythagorean theorem: $r = \sqrt{x^2 + y^2}$

2. Polar Angle (θ): This angle is between the positive x-axis and the line that connects the origin to the point. We find it using the arctangent function: $\theta = \tan^{-1}\left(\frac{y}{x}\right)$

Quadrant Considerations

To find the angle θ correctly, we need to know which quadrant the point $(x, y)$ is in:

• Quadrant I: If $x > 0$ and $y > 0$, then $\theta = \tan^{-1}\left(\frac{y}{x}\right)$.

• Quadrant II: If $x < 0$ and $y > 0$, then $\theta = \tan^{-1}\left(\frac{y}{x}\right) + \pi$.

• Quadrant III: If $x < 0$ and $y < 0$, then $\theta = \tan^{-1}\left(\frac{y}{x}\right) + \pi$.

• Quadrant IV: If $x > 0$ and $y < 0$, then $\theta = \tan^{-1}\left(\frac{y}{x}\right) + 2\pi$.

• On Axes:

  • If $x = 0$ and $y > 0$, then $\theta = \frac{\pi}{2}$.
  • If $x = 0$ and $y < 0$, then $\theta = \frac{3\pi}{2}$.
  • If $x > 0$ and $y = 0$, then $\theta = 0$.
  • If $x < 0$ and $y = 0$, then $\theta = \pi$.

Final Conversion Formula

To wrap it all up, converting from Cartesian coordinates to polar coordinates involves these formulas:

• $r = \sqrt{x^2 + y^2}$
• $\theta = \tan^{-1}\left(\frac{y}{x}\right) + k\pi \text{ (where $k$ depends on the quadrant)}$

Example

Let’s see how to do this with an example. Suppose we want to convert the Cartesian point $(3, 4)$ into polar coordinates.

1. Calculate r: $r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

2. Calculate θ: Since $(3, 4)$ is in Quadrant I (both x and y are positive): $\theta = \tan^{-1}\left(\frac{4}{3}\right)$

When we use a calculator, we find: $\theta \approx 0.927 \text{ radians}$

So, the polar coordinates of the point are: $(5, 0.927)$

Summary

To sum up, changing Cartesian coordinates to polar coordinates means:

• Finding the distance r from the origin using the square root of the sum of the squares of the Cartesian coordinates.
• Finding the angle θ using the arctangent of the ratio of the y-coordinate to the x-coordinate, taking into account which quadrant the point is in.

By learning these steps, you can easily switch between Cartesian and polar coordinates. This skill helps you understand math better, especially in geometry and calculus.