-in-polar-coordinates.jpeg "(-1 1) In Polar Coordinates")
Understanding (-1, 1) in Polar Coordinates
The point (-1, 1) in the Cartesian coordinate system represents a point that is 1 unit to the left of the origin and 1 unit above the origin. However, the same point can be expressed in polar coordinates using a different system. Let's explore how to convert this point and understand its representation in polar coordinates.
Polar Coordinates Explained
Polar coordinates use a different framework to locate points on a plane. Instead of using horizontal (x) and vertical (y) axes, polar coordinates use:
- Radius (r): The distance from the origin to the point.
- Angle (θ): The angle measured counter-clockwise from the positive x-axis to the line connecting the origin and the point.
Converting (-1, 1) to Polar Coordinates
To convert (-1, 1) to polar coordinates, we need to find its radius and angle:
1. Finding the Radius (r)
We can use the Pythagorean theorem to find the radius:
- r² = x² + y²
- r² = (-1)² + (1)²
- r² = 2
- r = √2
2. Finding the Angle (θ)
We can use trigonometry to find the angle:
- tan(θ) = y/x
- tan(θ) = 1/-1
- θ = 135° (or 3π/4 radians)
Since (-1, 1) lies in the second quadrant, the angle is greater than 90° and less than 180°.
Representing (-1, 1) in Polar Coordinates
Therefore, the polar coordinates of the point (-1, 1) are (√2, 135°) or (√2, 3π/4).
Important Notes
- The angle in polar coordinates can be expressed in degrees or radians.
- The angle can be positive or negative depending on the direction of rotation.
- The radius is always positive.
- The same point can be represented by multiple sets of polar coordinates due to the periodic nature of angles.
Understanding the conversion process from Cartesian to polar coordinates allows us to represent points in a different coordinate system, offering a different perspective on their location and relationships. This is particularly useful in various fields like physics, engineering, and mathematics.