-in-polar-coordinates.jpeg "(1 1) In Polar Coordinates")
The Point (1, 1) in Polar Coordinates
The point (1, 1) in Cartesian coordinates represents a location one unit to the right and one unit up from the origin. This point can also be expressed in polar coordinates, which use a different system for describing location.
Understanding Polar Coordinates
Polar coordinates use two components:
- Radius (r): The distance from the origin to the point.
- Angle (θ): The angle between the positive x-axis and the line connecting the origin to the point, measured counter-clockwise.
Converting (1, 1) to Polar Coordinates
To find the polar coordinates of (1, 1), we need to determine its radius and angle:
1. Finding the Radius (r):
We can use the Pythagorean theorem to find the distance from the origin to (1, 1):
r = √(1² + 1²) = √2
2. Finding the Angle (θ):
The point (1, 1) lies in the first quadrant, so the angle will be between 0° and 90°. We can use the arctangent function (tan⁻¹) to find the angle:
θ = tan⁻¹(1/1) = tan⁻¹(1) = 45°
Therefore, the polar coordinates of (1, 1) are (√2, 45°).
Note:
- There are infinitely many ways to express a point in polar coordinates because adding 360° (or any multiple of 360°) to the angle results in the same point. For example, (√2, 405°) and (√2, 765°) also represent the point (1, 1) in polar coordinates.
- Angles in polar coordinates are typically expressed in radians rather than degrees. To convert 45° to radians, multiply by π/180: 45° * (π/180) = π/4 radians. So, the polar coordinates of (1, 1) can also be written as (√2, π/4).