To find all polar coordinates of a point given in Cartesian coordinates, we use the relationship between Cartesian coordinates (x, y) and polar coordinates (r, θ). Here, the Cartesian coordinates are (9, 75).
1. Calculating the radius (r): The radius in polar coordinates can be calculated using the formula:
r = √(x² + y²)
Substituting the values:
r = √(9² + 75²) = √(81 + 5625) = √5706 ≈ 75.5
2. Calculating the angle (θ): The angle can be determined using the arctangent function:
θ = arctan(y/x)
Now substituting our values:
θ = arctan(75/9) ≈ 82.87 degrees
3. Finding all polar coordinates: In polar coordinates, you can describe a point using the radius and angle. However, since angles can be represented in multiple ways, we can express the angle in different forms:
- Basic form: (r, θ) = (75.5, 82.87 degrees)
- Adding full rotations: (r, θ + 360n), where n is any integer. For example, (75.5, 82.87 + 360*1) = (75.5, 442.87 degrees)
- Negative radius: In polar coordinates, a negative r indicates a direction opposite to θ. Thus, we can also express this as (-r, θ + 180 degrees). For instance, (-75.5, 82.87 + 180) = (-75.5, 262.87 degrees)
4. Summary: Therefore, the polar coordinates of the point P(9, 75) include:
- (75.5, 82.87 degrees)
- (75.5, 442.87 degrees)
- (-75.5, 262.87 degrees)
Each of these coordinates represents the same point in polar form, showcasing the beauty and flexibility of polar coordinates.