-in-polar-coordinates.jpeg "(3 3) In Polar Coordinates")
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:
-
Find the distance (r):
- Use the Pythagorean theorem: r = √(x² + y²) = √(3² + 3²) = √18 = 3√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.