To find two pairs of polar coordinates for the point (3, 3), we first need to understand the relationship between Cartesian coordinates and polar coordinates.
In polar coordinates, a point is defined by a radius (r) and an angle (θ). The conversion from Cartesian to polar coordinates uses the formulas:
- r = √(x² + y²)
- θ = arctan(y/x) (in radians, and convert to degrees if necessary)
For the point (3, 3):
- Calculate the radius (r):
- r = √(3² + 3²) = √(9 + 9) = √18 = 3√2 ≈ 4.24
- Calculate the angle (θ):
- θ = arctan(3/3) = arctan(1) = 45°
This gives us the first pair of polar coordinates:
- Pair 1: (r, θ) = (4.24, 45°)
Now, let’s find the second pair of polar coordinates. In polar coordinates, angles can be represented in multiple ways. Adding or subtracting 360° will give us equivalent angles. Hence, we can also represent our angle by adding 360° to our first angle:
- θ = 45° + 360° = 405°
This results in another valid representation:
- Pair 2: (r, θ) = (4.24, 405°)
In summary, the two pairs of polar coordinates for the point (3, 3) are:
- Pair 1: (4.24, 45°)
- Pair 2: (4.24, 405°)
Both pairs accurately represent the same point in polar coordinates, illustrating the flexibility of polar notation.