Polar Coordinate System | Generated by AI
The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. This is in contrast to the Cartesian coordinate system, which uses x and y coordinates to define a point.
In polar coordinates, a point is represented as \((r, \theta)\), where:
- \(r\) is the radial distance from the origin (the reference point).
- \(\theta\) is the angular coordinate, usually measured in radians from the positive x-axis (the reference direction).
Examples of Polar Coordinates
- Simple Point:
- Cartesian Coordinates: \((1, 0)\)
- Polar Coordinates: \((1, 0)\)
- Explanation: The point is 1 unit away from the origin along the positive x-axis.
- Point in the Plane:
- Cartesian Coordinates: \((0, 1)\)
- Polar Coordinates: \((1, \frac{\pi}{2})\)
- Explanation: The point is 1 unit away from the origin, at an angle of \(\frac{\pi}{2}\) radians (90 degrees) from the positive x-axis.
- Point in the Second Quadrant:
- Cartesian Coordinates: \((-1, 1)\)
- Polar Coordinates: \((\sqrt{2}, \frac{3\pi}{4})\)
- Explanation: The point is \(\sqrt{2}\) units away from the origin, at an angle of \(\frac{3\pi}{4}\) radians (135 degrees) from the positive x-axis.
- Point in the Third Quadrant:
- Cartesian Coordinates: \((-1, -1)\)
- Polar Coordinates: \((\sqrt{2}, \frac{5\pi}{4})\)
- Explanation: The point is \(\sqrt{2}\) units away from the origin, at an angle of \(\frac{5\pi}{4}\) radians (225 degrees) from the positive x-axis.
- Point in the Fourth Quadrant:
- Cartesian Coordinates: \((1, -1)\)
- Polar Coordinates: \((\sqrt{2}, \frac{7\pi}{4})\)
- Explanation: The point is \(\sqrt{2}\) units away from the origin, at an angle of \(\frac{7\pi}{4}\) radians (315 degrees) from the positive x-axis.
Conversion Between Cartesian and Polar Coordinates
-
From Cartesian to Polar: \[ r = \sqrt{x^2 + y^2} \] \[ \theta = \arctan\left(\frac{y}{x}\right) \]
-
From Polar to Cartesian: \[ x = r \cdot \cos(\theta) \] \[ y = r \cdot \sin(\theta) \]
Polar coordinates are particularly useful in situations involving circular symmetry, such as in physics and engineering problems involving rotation or circular motion.