To determine the polar coordinates of the point (3, 3), we first need to convert these Cartesian coordinates to polar coordinates. In polar coordinates, a point is represented by the radius (r) and the angle (θ).
1. **Calculating the Radius (r)**:
The radius can be calculated using the formula:
r = √(x2 + y2)
where x = 3 and y = 3.
Thus:
r = √(32 + 32)
r = √(9 + 9)
r = √18
r = 3√2 ≈ 4.24
2. **Calculating the Angle (θ)**:
The angle can be calculated using the arctangent function:
θ = tan-1(y/x)
Substituting the values:
θ = tan-1(3/3)
θ = tan-1(1)
θ = 45°
3. **Generating Polar Coordinate Pairs**:
Now that we have r and θ, we can express the coordinates. The primary pair of polar coordinates for the point (3, 3) is:
- (r, θ) = (4.24, 45°)
Next, we can also consider the additional pairs using the angle adjustments. Polar coordinates can also represent a point using angles shifted by 180 degrees. Therefore:
- (r, θ + 360°) = (4.24, 405°)
- (r, θ + 180°) = (4.24, 225°)
4. **Considering the Given Angles**:
The angles provided (0°, 8°, and 360°) can also be utilized to express the first coordinate:
- (r, θ + 0°) = (4.24, 45°)
- (r, θ + 8°) = (4.24, 53°)
- (r, θ + 360°) = (4.24, 405°)
Thus, the two pairs of polar coordinates are:
- (4.24, 45°)
- (4.24, 225°)
In summary, the pairs of polar coordinates for the point (3, 3) are approximately:
- (4.24, 45°)
- (4.24, 225°)