How do you convert Cartesian coordinates (x, y) to polar coordinates (r, θ)?

To convert Cartesian coordinates (x, y) to polar coordinates (r, θ), you need to understand how these two coordinate systems relate to each other.

1. **Calculate the Radius (r)**: The radius is the distance from the origin (0, 0) to the point (x, y). It is calculated using the Pythagorean theorem:

r = √(x² + y²)

2. **Calculate the Angle (θ)**: The angle θ is the inclination of the line connecting the origin to the point (x, y) with respect to the positive x-axis. It is calculated using the tangent function:

θ = arctan(y / x)

Be mindful of the signs of x and y to determine the correct quadrant for θ:

• If x > 0 and y > 0, then θ is in the first quadrant (0 < θ < 90 degrees).
• If x < 0 and y > 0, then θ is in the second quadrant (90 < θ < 180 degrees).
• If x < 0 and y < 0, then θ is in the third quadrant (180 < θ < 270 degrees).
• If x > 0 and y < 0, then θ is in the fourth quadrant (270 < θ < 360 degrees).

3. **Final Representation**: Once you have calculated r and θ, the polar coordinates can be represented as (r, θ).

For example, if you have the Cartesian coordinates (3, 4):

• Calculate r:
• r = √(3² + 4²) = √(9 + 16) = √25 = 5
• Calculate θ:
• θ = arctan(4 / 3) = 53.13 degrees

Thus, the polar coordinates would be (5, 53.13 degrees).