Polar-Rectangular Conversions — Mathematics Knowledge System

Overview

Every point in the plane can be located either by rectangular coordinates (x, y) measured along perpendicular axes, or by polar coordinates (r, θ) giving a distance from the origin and an angle from the positive x-axis. Because a polar point sits at the vertex of a right triangle formed by dropping a perpendicular to the x-axis, the unit-circle definitions of sine and cosine translate directly between the two systems: x = r cosθ and y = r sinθ convert polar to rectangular, while r = √(x²+y²) and tanθ = y/x (with quadrant care) convert rectangular to polar. These conversions are essential for working with polar equations, complex numbers in polar form, and any physical situation — orbits, radar, robotics — where distance-and-angle data must be combined with Cartesian data.

Intuition

Drop a perpendicular from a polar point (r, θ) to the x-axis. This creates a right triangle with hypotenuse r, and the acute angle at the origin is θ (measured appropriately for the quadrant). By the unit-circle definitions, cosθ is the ratio of the adjacent side to the hypotenuse and sinθ is the ratio of the opposite side to the hypotenuse — so the adjacent side (the x-coordinate) is r cosθ and the opposite side (the y-coordinate) is r sinθ. Going the other direction, the Pythagorean theorem on that same right triangle gives r = √(x²+y²), and the tangent ratio y/x recovers θ, though you must check which quadrant (x, y) lies in since tanθ alone cannot distinguish, say, θ from θ+π.

Formal Definition

Definition

For a point with polar coordinates (r, θ) and rectangular coordinates (x, y):

x = r\cos\theta, \qquad y = r\sin\theta

Polar to rectangular

r = \sqrt{x^2 + y^2}, \qquad \tan\theta = \frac{y}{x} \ (x \ne 0)

Rectangular to polar

Worked Examples

Apply x = r cosθ with r=4, θ=30°.
$x = 4\cos(30°) = 4\cdot\frac{\sqrt3}{2} = 2\sqrt3$
Apply y = r sinθ.
$y = 4\sin(30°) = 4\cdot\frac{1}{2} = 2$

Answer: Rectangular coordinates: (2√3, 2) ≈ (3.4641, 2).

Practice Problems

Difficulty 3/10

Convert the polar point (r, θ) = (6, 120°) to rectangular coordinates.

Difficulty 4/10

Convert the rectangular point (−3, −3) to polar coordinates with 0 ≤ θ < 360°.

Difficulty 6/10

A radar station detects an aircraft at distance r = 10 km and bearing θ = 150° from due east (measured counterclockwise). Find the aircraft's rectangular position (x east, y north) relative to the station.

Quiz

Summary

x = r cosθ and y = r sinθ convert polar coordinates to rectangular, directly from the unit-circle definitions of sine and cosine.
r = √(x²+y²) and tanθ = y/x convert rectangular to polar, with the quadrant of (x, y) determining the correct θ.
These conversions underlie complex numbers in polar form and any application (radar, orbital mechanics, robotics) mixing distance-angle and Cartesian data.

References

WebsiteWikipedia — Polar coordinate system