-polar-coordinates.jpeg "(-3 4) Polar Coordinates")
Converting Rectangular Coordinates to Polar Coordinates
The point (-3, 4) represents a point in the rectangular coordinate system. To convert this point to polar coordinates, we need to find its distance from the origin (radius, r) and the angle it makes with the positive x-axis (angle, θ).
Finding the radius (r):
We can find r using the Pythagorean theorem:
r = √((-3)² + (4)²) = √(9 + 16) = √25 = 5
Finding the angle (θ):
We can use the arctangent function to find the angle:
θ = arctan(y/x) = arctan(4/-3) ≈ -0.93 radians
However, the angle we calculated is in the fourth quadrant, while the point (-3, 4) lies in the second quadrant. To get the correct angle, we need to add π radians:
θ = -0.93 + π ≈ 2.21 radians
Therefore, the polar coordinates of the point (-3, 4) are (5, 2.21 radians).
Understanding Polar Coordinates
Polar coordinates use a different system to represent points in space. Instead of using horizontal and vertical axes, polar coordinates use a distance from the origin (r) and an angle from the positive x-axis (θ).
Advantages of Polar Coordinates:
- Simplifying certain geometric shapes: Circles, spirals, and other shapes that are complex to describe in rectangular coordinates can be easily expressed in polar coordinates.
- Representing periodic phenomena: Polar coordinates are often used to represent phenomena that are periodic in nature, such as the movement of planets or the oscillation of a pendulum.
Remember that polar coordinates are not unique. Adding multiples of 2π to the angle will result in the same point. For example, (5, 2.21) is the same point as (5, 2.21 + 2π) and (5, 2.21 + 4π).