-in-polar-coordinates.jpeg "(-6 6) In Polar Coordinates")
Converting (-6, 6) to Polar Coordinates
To convert a point from rectangular coordinates (x, y) to polar coordinates (r, θ), we use the following formulas:
- r = √(x² + y²)
- θ = arctan(y/x) (with adjustments based on the quadrant of the point)
Let's apply these formulas to the point (-6, 6):
1. Calculate r:
- r = √((-6)² + 6²)
- r = √(36 + 36)
- r = √72 = 6√2
2. Calculate θ:
- θ = arctan(6/-6)
- θ = arctan(-1)
- θ = -π/4
However, the point (-6, 6) lies in the second quadrant, where the angle should be between π/2 and π. Therefore, we need to add π to our calculated θ:
- θ = -π/4 + π
- θ = 3π/4
Therefore, the polar coordinates of (-6, 6) are (6√2, 3π/4).
Important Note: It is crucial to consider the quadrant of the point when determining the angle θ. Using only the arctan function can lead to incorrect results.