Relation between cartesian and polar coordinate

Relation Between Cartesian and Polar Coordinates

In coordinate geometry, a point in the plane can be represented in two common ways: Cartesian coordinates and Polar coordinates.

1. Cartesian Coordinates

A point P is represented as (x, y), where:

• x is the horizontal distance from the origin (O) to point P.
• y represents the vertical displacement of point P from the origin (O).

2. Polar Coordinates

A point P is represented as (r, θ), where:

• r is the distance from the origin (O) to point P.
• θ (theta) is the angle made by the line OP with the positive x-axis.

Geometric Representation

Draw a point P in the coordinate plane. From point P, drop a perpendicular to the x-axis and label the foot of the perpendicular as Q. Let O be the origin. Then:

• OP = r, which denotes the straight-line distance between the origin (O) and point P.
• ∠POQ = θ (angle between OP and x-axis)
• In triangle POQ:
  • OQ = x (base)
  • PQ = y (height)
  • OP = r (hypotenuse)

Relation Between Cartesian and Polar Coordinates

Using trigonometry in triangle POQ:

• x = r cos θ
• y = r sin θ

Also, by Pythagoras theorem:

• r = √(x² + y²)
• θ = tan⁻¹(y / x), considering the appropriate quadrant for the angle.

Conversion Formulas

1. Cartesian to Polar:

• r = √(x² + y²)
• θ = tan⁻¹(y / x)

2. Polar to Cartesian:

• x = r cos θ
• y = r sin θ

Example

Suppose a point is represented in Cartesian coordinates as (3, 4)

• r = √(3² + 4²) = √25 = 5
• θ = tan⁻¹(4/3) ≈ 53.13°
• So, Polar form = (5, 53.13°)

Given a point in Polar form: (5, 53.13°)

• x = 5 cos(53.13°) ≈ 3
• y = 5 sin(53.13°) ≈ 4
• So, Cartesian form = (3, 4)