-in-polar-coordinates.jpeg "(5 5) In Polar Coordinates")
Understanding (5, 5) in Polar Coordinates
Polar coordinates offer a different way to represent points in a two-dimensional space compared to the familiar Cartesian coordinates (x, y). Instead of using horizontal and vertical distances, polar coordinates use a radius (distance from the origin) and an angle (measured counter-clockwise from the positive x-axis).
Converting from Cartesian to Polar
Let's consider the point (5, 5) in Cartesian coordinates. To convert this to polar coordinates, we need to find the radius (r) and the angle (θ).
1. Finding the Radius:
- The radius is simply the distance from the origin to the point (5, 5). We can use the Pythagorean theorem:
- r² = x² + y²
- r² = 5² + 5²
- r = √50 = 5√2
2. Finding the Angle:
- The angle can be found using the arctangent function:
- θ = arctan(y/x)
- θ = arctan(5/5)
- θ = arctan(1)
- θ = 45° (or π/4 radians)
Therefore, the polar coordinates of (5, 5) are (5√2, 45°) or (5√2, π/4).
Visualizing the Point
Imagine a circle centered at the origin with a radius of 5√2. The point (5, 5) lies on this circle, 45° counter-clockwise from the positive x-axis. This is how the point (5, 5) is represented in polar coordinates.
Key Points to Remember
- Polar coordinates are represented as (r, θ) where r is the radius and θ is the angle.
- The angle θ is measured counter-clockwise from the positive x-axis.
- The same point can have multiple polar representations due to the periodic nature of angles. For example, (5√2, 45°) is equivalent to (5√2, 405°) because adding 360° (or 2π radians) to the angle doesn't change the point's location.
Understanding the conversion between Cartesian and polar coordinates helps us analyze points and functions in different contexts, particularly when dealing with circular or rotational concepts.