Cartesian to polar equations (Circle, line)

Cartesian to polar equations: The Cartesian coordinate system and the Polar coordinate system are two different ways of representing points in a plane. In this blog, we’ll explain how to convert Cartesian coordinates (x, y) to Polar coordinates (r, θ) and vice versa, with examples to help understand.

1. Cartesian to Polar Conversion Examples

Quadrant I:

• Cartesian coordinates: (3,4)
  • Find r:

• Find θ:

Polar coordinates: (r=5,θ=0.93) or (r=5,θ=53.13∘)

Quadrant II:

• Cartesian coordinates: (−3,4)
  • Find r:

• Find θ:

Since the point is in the second quadrant, we need to add 𝜋 (or 180°) to achieve the correct angle.

Polar coordinates: (r=5,θ=2.21) or (r=5,θ=126.87∘)

Quadrant III:

• Cartesian coordinates(−3,−4)

• Find r

• Find θ:

Since the point is in the third quadrant, we need to add 𝜋 (or 180°) to the angle.

Polar coordinates: (r=5,θ=2.21) or (r=5,θ=126.87∘)

Quadrant IV:

• Cartesian coordinates: $(3,-4)$
  • Find r

• Find θ

As the point is in the fourth quadrant, the angle is already correct:

• Polar coordinates: (r=5,θ=−0.93) or (r=5,θ=−53.13∘)

The following are examples of several types of Cartesian to polar equations and their polar transformations, such as straight lines , conic sections, and others.

1.Equation: y=3x+1 (Line)

Step 1: substitute for x and y.

Substitute into the equation y=3x+1

Step 2: Simplify Divide both sides by 𝑟 if 𝑟 ≠ 0 and r = 0.

Step 3: Solve for r

In polar form, the equation  y = 3x+1 becomes:

This equation shows a straight line in polar coordinates.

2. Equation: 𝑥^2 + 𝑦^ 2 = 16 (circle).

Step 1: Substitute for x and y

Substitute into the equation  x ^2 +y^ 2 = 16x:

Step 2: Simplify Divide both sides by  r (assuming 𝑟 ≠0 and r=0):

Polar Form: The equation x^2 + y^2 = 16x in polar form is:

This represents a circle with a radius of 8, centered at (8,0) in Cartesian coordinates.To learn more, check this link.