Polar to Rectangular Calculator – Convert Coordinates
Convert polar coordinates to rectangular (Cartesian) form
How to Use
- Enter the radius (r) value
- Enter the angle (θ) value
- Select whether the angle is in degrees or radians
- Click calculate to convert to rectangular coordinates
- View the x and y coordinates and complex number form
What are Polar Coordinates?
Polar coordinates represent a point using a distance from the origin (radius r) and an angle from the positive x-axis (θ). This system is particularly useful for describing circular and spiral patterns.
A point in polar form is written as (r, θ), where r is the radial distance and θ (theta) is the angular coordinate measured counterclockwise from the positive x-axis.
Conversion Formulas
To convert from polar (r, θ) to rectangular (x, y) coordinates:
- x = r × cos(θ)
- y = r × sin(θ)
To convert from rectangular (x, y) to polar (r, θ) coordinates:
- r = √(x² + y²)
- θ = arctan(y/x) (with quadrant adjustment)
Connection to Complex Numbers
Polar coordinates are closely related to complex numbers. A complex number z = x + yi can be written in polar form as z = r(cos θ + i sin θ) or using Euler's formula as z = re^(iθ).
This connection makes polar form especially useful for multiplication and division of complex numbers, as well as finding roots.
Common Angle Conversions
| Degrees | Radians | cos(θ) | sin(θ) |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | π/6 | √3/2 | 1/2 |
| 45° | π/4 | √2/2 | √2/2 |
| 60° | π/3 | 1/2 | √3/2 |
| 90° | π/2 | 0 | 1 |
| 180° | π | -1 | 0 |
| 270° | 3π/2 | 0 | -1 |
Applications
- Navigation and GPS systems
- Radar and sonar systems
- Robotics and motion planning
- Signal processing and Fourier analysis
- Physics: circular motion and waves
- Computer graphics and game development
- Electrical engineering: AC circuits
Frequently Asked Questions
- When should I use polar coordinates instead of rectangular?
- Polar coordinates are ideal when dealing with circular or rotational problems, such as describing orbits, spirals, or any situation where distance from a center point and angle are more natural measurements than x and y positions.
- Can the radius be negative?
- Yes, a negative radius means the point is in the opposite direction. The point (-r, θ) is the same as (r, θ + 180°). This convention is sometimes used in mathematics but can be confusing, so positive radii are more common.
- How do I convert degrees to radians?
- Multiply degrees by π/180. For example, 90° = 90 × π/180 = π/2 radians. Conversely, multiply radians by 180/π to get degrees.
- What's the relationship between polar form and complex numbers?
- A complex number x + yi corresponds to the point (x, y) in rectangular coordinates, which equals (r, θ) in polar form where r = √(x² + y²) and θ = arctan(y/x). This is written as r∠θ or r·e^(iθ).