-polar-coordinates.jpeg "(3 5pi/6) Polar Coordinates")
Understanding Polar Coordinates: (3, 5π/6)
Polar coordinates are a different way to represent points in a plane compared to the standard Cartesian (x, y) system. They use a distance from the origin (r) and an angle from the positive x-axis (θ).
Let's explore the point (3, 5π/6) in polar coordinates:
Breaking Down the Coordinates:
- r = 3: This means the point is located 3 units away from the origin.
- θ = 5π/6: This angle is measured counterclockwise from the positive x-axis.
Visualizing the Point:
- Start at the origin: This is the center of your coordinate system.
- Rotate counterclockwise: Rotate the angle 5π/6 (150 degrees) from the positive x-axis.
- Extend the radius: From the point you landed on after rotating, extend a line outward 3 units.
The point where the line ends is the location of (3, 5π/6) in polar coordinates.
Converting to Cartesian Coordinates:
To understand the relationship between polar and Cartesian coordinates, we can use the following conversion formulas:
- x = r cos(θ)
- y = r sin(θ)
Applying these to our point:
- x = 3 * cos(5π/6) = -3√3 / 2
- y = 3 * sin(5π/6) = 3 / 2
Therefore, the Cartesian coordinates of (3, 5π/6) are (-3√3 / 2, 3 / 2).
Understanding the Relationship:
Polar coordinates offer a valuable way to describe points, especially in scenarios involving circular or rotational symmetry. They can be particularly useful when dealing with curves defined by their distance from a central point and their angle relative to a fixed direction.