-polar-coordinates.jpeg "(-4 4) Polar Coordinates")
Understanding Polar Coordinates: The Case of (-4, 4)
Polar coordinates offer a different way to represent points in a plane compared to the familiar Cartesian (x, y) system. Instead of using horizontal and vertical distances, polar coordinates use distance from the origin (r) and angle from the positive x-axis (θ).
Let's break down how to find the polar coordinates for the point (-4, 4) in the Cartesian plane:
1. Finding the Distance (r)
The distance from the origin is calculated using the Pythagorean theorem:
r = √(x² + y²) = √((-4)² + 4²) = √(16 + 16) = √32 = 4√2
2. Finding the Angle (θ)
To find the angle, we need to consider the quadrant where the point lies. (-4, 4) is in the second quadrant. We can use the arctangent function (tan⁻¹) to find the angle:
θ = tan⁻¹(y/x) = tan⁻¹(4/-4) = tan⁻¹(-1)
The arctangent of -1 is -45°, but this is the angle measured from the positive x-axis in the fourth quadrant. Since our point is in the second quadrant, we need to add 180°:
θ = -45° + 180° = 135°
Expressing the Polar Coordinates
Therefore, the polar coordinates for (-4, 4) are (4√2, 135°).
Additional Notes
- Polar coordinates are not unique. Adding multiples of 360° to the angle will result in the same point. For example, (4√2, 135°) is equivalent to (4√2, 495°) or (4√2, -225°).
- The angle θ can be expressed in radians, where 180° equals π radians. So, 135° in radians is (3π/4) radians. Therefore, the polar coordinates in radians would be (4√2, 3π/4).
By understanding how to convert between Cartesian and polar coordinates, you gain a powerful tool to represent points and analyze geometric relationships in a different light.