(3 3) In Polar Coordinates

Jasonbradley

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Jun 16, 2024

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Understanding (3, 3) in Polar Coordinates

Polar coordinates provide a different way to represent points in a plane compared to the traditional Cartesian coordinates (x, y). Instead of using horizontal and vertical distances, polar coordinates use distance from the origin (r) and angle from the positive x-axis (θ).

Converting Cartesian to Polar Coordinates

To convert the point (3, 3) from Cartesian to polar coordinates, we follow these steps:

1. Find the distance (r):

   • Use the Pythagorean theorem: r = √(x² + y²) = √(3² + 3²) = √18 = 3√2

2. Find the angle (θ):

   • Use the arctangent function: θ = arctan(y/x) = arctan(3/3) = arctan(1) = 45°

   • Since the point (3, 3) lies in the first quadrant, the angle is 45°

Therefore, the polar coordinates of (3, 3) are (3√2, 45°).

Visualizing the Point in Polar Coordinates

Imagine a circle with a radius of 3√2 centered at the origin. Now, imagine a line segment from the origin extending to a point on the circle that forms an angle of 45° with the positive x-axis. The endpoint of this line segment represents the point (3√2, 45°) in polar coordinates.

Key Takeaways

• Polar coordinates provide a different perspective on representing points in a plane.
• Conversion from Cartesian to polar coordinates involves finding the distance from the origin (r) and the angle from the positive x-axis (θ).
• The angle (θ) needs to be adjusted based on the quadrant where the point lies.
• Visualizing polar coordinates can help understand the relationship between Cartesian and polar representations.