(-3 4) To Polar Coordinates

3 min read Jun 16, 2024

Converting Rectangular Coordinates to Polar Coordinates

Converting rectangular coordinates to polar coordinates is a common process in mathematics, particularly when dealing with trigonometric functions and complex numbers. This article will guide you through the process of converting the rectangular coordinate (-3, 4) to its equivalent polar coordinates.

Understanding Polar Coordinates

Polar coordinates are a different way of representing points in a two-dimensional plane. Instead of using horizontal (x) and vertical (y) distances, polar coordinates use a radius (r) and an angle (θ).

Radius (r): This represents the distance from the origin (0, 0) to the point. It is always a non-negative value.
Angle (θ): This represents the angle between the positive x-axis and the line connecting the origin to the point. It is measured counterclockwise from the positive x-axis.

Converting Rectangular to Polar Coordinates

To convert rectangular coordinates (x, y) to polar coordinates (r, θ), we use the following formulas:

1. Finding the radius (r):

r = √(x² + y²)

2. Finding the angle (θ):

θ = arctan(y/x)

Important Note: The angle θ needs to be adjusted based on the quadrant of the point.

If the point is in Quadrant I or IV, the calculated angle is correct.
If the point is in Quadrant II, add 180° (or π radians) to the calculated angle.
If the point is in Quadrant III, add 180° (or π radians) to the calculated angle.

Applying the Formulas to (-3, 4)

Finding the radius (r):

r = √((-3)² + 4²) = √(9 + 16) = √25 = 5

Finding the angle (θ):
```
θ = arctan(4/-3) ≈ -53.13°
```
Since the point (-3, 4) lies in Quadrant II, we need to add 180° to the calculated angle:
```
θ ≈ -53.13° + 180° = 126.87°
```

Result

Therefore, the polar coordinates of (-3, 4) are approximately (5, 126.87°).

Remember: Angles can be expressed in degrees or radians. In this example, we used degrees.