(-3 3) To Polar Form

2 min read Jun 16, 2024
(-3 3) To Polar Form

Converting Rectangular Coordinates (-3, 3) to Polar Form

This article will guide you through the process of converting rectangular coordinates (-3, 3) to their equivalent polar form.

Understanding Rectangular and Polar Coordinates

  • Rectangular Coordinates: A point is represented by its horizontal distance (x-coordinate) and vertical distance (y-coordinate) from the origin. This is the familiar Cartesian coordinate system.
  • Polar Coordinates: A point is represented by its distance from the origin (radius, r) and the angle (theta, θ) it makes with the positive x-axis.

Formula for Conversion

To convert from rectangular coordinates (x, y) to polar coordinates (r, θ):

  • Radius (r):
    • r = √(x² + y²)
  • Angle (θ):
    • θ = arctan(y/x)

Applying the Formula to (-3, 3)

  1. Calculate the radius (r):

    • r = √((-3)² + (3)²)
    • r = √(9 + 9)
    • r = √18
    • r = 3√2
  2. Calculate the angle (θ):

    • θ = arctan(3/-3)
    • θ = arctan(-1)

Since (-3, 3) lies in the second quadrant, we need to adjust the angle. The arctan(-1) gives us -45°, but the angle in the second quadrant is 135°.

Therefore, the polar coordinates of (-3, 3) are (3√2, 135°).

Conclusion

By applying the formulas and understanding the relationship between rectangular and polar coordinates, we successfully converted the point (-3, 3) to its equivalent polar form (3√2, 135°).

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