)-in-polar-coordinates.jpeg "(3 3 Sqrt(3)) In Polar Coordinates")
Representing (3, 3√3) in Polar Coordinates
The point (3, 3√3) in rectangular coordinates represents a point 3 units to the right of the origin and 3√3 units above the origin. We can convert this to polar coordinates (r, θ) using the following formulas:
- r = √(x² + y²): This calculates the distance from the origin to the point.
- θ = tan⁻¹(y/x): This determines the angle from the positive x-axis to the line connecting the origin and the point.
Let's calculate:
-
r = √(3² + (3√3)²) = √(9 + 27) = √36 = 6
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θ = tan⁻¹((3√3)/3) = tan⁻¹(√3) = 60°
Therefore, the polar coordinate representation of (3, 3√3) is (6, 60°).
Important Note: When using the arctangent function (tan⁻¹) to find θ, it's crucial to consider the quadrant of the original rectangular point. In this case, (3, 3√3) lies in the first quadrant, and our calculated angle 60° is also in the first quadrant.
Understanding Polar Coordinates
Polar coordinates describe a point in terms of its distance from the origin (r) and the angle (θ) it makes with the positive x-axis. The angle is measured counterclockwise from the positive x-axis.
The polar coordinate representation of a point is not unique. For instance, (6, 60°) and (6, 420°) represent the same point since adding 360° (or any multiple of 360°) to the angle doesn't change the location of the point.