-to-polar-coordinates.jpeg "(-4 4) To Polar Coordinates")
Converting Rectangular Coordinates (-4, 4) to Polar Coordinates
This article will guide you through the process of converting rectangular coordinates (-4, 4) to polar coordinates.
Understanding Rectangular and Polar Coordinates
Rectangular coordinates are represented as (x, y), where x is the horizontal distance from the origin and y is the vertical distance from the origin.
Polar coordinates are represented as (r, θ), where r is the distance from the origin to the point and θ is the angle measured counterclockwise from the positive x-axis.
Conversion Formulas
To convert from rectangular coordinates (x, y) to polar coordinates (r, θ), we use the following formulas:
- r = √(x² + y²)
- θ = arctan(y/x) (taking into account the quadrant of the point)
Applying the Formulas
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Finding r:
- r = √((-4)² + (4)²)
- r = √(16 + 16)
- r = √32
- r = 4√2
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Finding θ:
- θ = arctan(4/-4)
- θ = arctan(-1)
Since the point (-4, 4) lies in the second quadrant, we need to add π to the arctan value to get the correct angle.
- θ = arctan(-1) + π
- θ = -π/4 + π
- θ = 3π/4
Result
Therefore, the polar coordinates of (-4, 4) are (4√2, 3π/4).
Conclusion
By applying the conversion formulas, we successfully converted the rectangular coordinates (-4, 4) to the polar coordinates (4√2, 3π/4). Remember to pay attention to the quadrant of the point when calculating the angle θ.