Polar to Rectangular Converter

Convert polar coordinates (r, θ) to rectangular coordinates (x, y) with step-by-step solutions.

Enter Polar Coordinates

Result

Rectangular Coordinates
(3.5355, 3.5355)
(x, y)
x-coordinate 3.535534
y-coordinate 3.535534
Angle in radians 0.785398
Angle in degrees 45

Step-by-Step Solution

x = r cos(θ), y = r sin(θ)

Understanding Polar to Rectangular Conversion

Polar coordinates represent a point in the plane using a distance from the origin (r) and an angle from the positive x-axis (θ). Rectangular (Cartesian) coordinates represent a point using horizontal (x) and vertical (y) distances from the origin. Converting between these systems is fundamental in mathematics, physics, and engineering.

Conversion Formulas

Polar to Rectangular

Convert from (r, θ) to (x, y) using trigonometric functions.

x = r cos(θ), y = r sin(θ)

Rectangular to Polar

Convert from (x, y) to (r, θ) using distance and inverse tangent.

r = sqrt(x² + y²), θ = atan2(y, x)

Degree-Radian Conversion

Angles can be measured in degrees or radians.

radians = degrees x π / 180

Why Use Different Coordinate Systems?

Polar coordinates are especially useful for problems involving circular or rotational symmetry. For example, describing the motion of a satellite orbiting Earth is much simpler in polar coordinates. Rectangular coordinates are preferred for problems involving straight lines, grids, and most algebraic operations.

Common Applications

  • Physics: Describing circular motion, wave patterns, and electromagnetic fields.
  • Engineering: Radar systems, antenna design, and signal processing.
  • Navigation: GPS coordinates and bearing-based navigation.
  • Computer Graphics: Rotating objects, drawing arcs and spirals.

Special Angle Values

Some angles produce well-known coordinate values that are worth memorizing:

  • θ = 0 degrees: (r, 0) -- point on the positive x-axis
  • θ = 90 degrees: (0, r) -- point on the positive y-axis
  • θ = 180 degrees: (-r, 0) -- point on the negative x-axis
  • θ = 270 degrees: (0, -r) -- point on the negative y-axis
  • θ = 45 degrees: (r/sqrt(2), r/sqrt(2)) -- diagonal in first quadrant

Tips for Accurate Conversions

  • Always verify whether the angle is in degrees or radians before converting.
  • Remember that negative values of r reflect the point through the origin.
  • For angles greater than 360 degrees, reduce by subtracting multiples of 360.
  • The rectangular form is unique, but the polar form has infinitely many representations.