-to-polar.jpeg "(-3 4) To Polar")
Converting Rectangular Coordinates (-3, 4) to Polar Coordinates
This article will guide you through the process of converting rectangular coordinates (-3, 4) to polar coordinates.
Understanding Rectangular and Polar Coordinates
Rectangular coordinates (also known as Cartesian coordinates) use a horizontal x-axis and a vertical y-axis to define a point's location on a plane. A point is represented as (x, y).
Polar coordinates, on the other hand, use a radius (r) and an angle (θ) to define a point's location. The radius represents the distance from the origin, and the angle represents the counterclockwise rotation from the positive x-axis. A point is represented as (r, θ).
Conversion Formulas
To convert from rectangular coordinates (x, y) to polar coordinates (r, θ), we use the following formulas:
- r = √(x² + y²): This formula calculates the distance from the origin (the radius).
- θ = arctan(y/x): This formula calculates the angle from the positive x-axis.
Converting (-3, 4)
Let's apply these formulas to our coordinates (-3, 4):
-
Find r:
- r = √((-3)² + (4)²)
- r = √(9 + 16)
- r = √25
- r = 5
-
Find θ:
- θ = arctan(4 / -3)
- θ ≈ -53.1°
Important Note: The arctan function gives you an angle in the range of -90° to 90°. Since our point (-3, 4) is in the second quadrant, we need to adjust the angle. We can add 180° to the angle to get the correct value:
- θ ≈ -53.1° + 180°
- θ ≈ 126.9°
The Result
Therefore, the polar coordinates of (-3, 4) are (5, 126.9°).
Visualizing the Conversion
Imagine drawing a right triangle with the origin as one vertex, (-3, 4) as another vertex, and the point (-3, 0) as the third vertex. The radius 'r' is the hypotenuse of this triangle, and the angle 'θ' is the angle between the positive x-axis and the hypotenuse.