To find all polar coordinates of the point P(9, 75), we begin with the Cartesian coordinates provided, where ‘9’ is the radius (r) and ’75’ is the angle (θ) measured in degrees.
In polar coordinates, any point can be represented as (r, θ), and points that are equivalent can be represented by adding or subtracting multiples of 360° from the angle θ. This accounts for the periodic nature of the circular system.
Given that we have r = 9 and θ = 75°, one possible representation in polar coordinates is (9, 75°).
To find other polar coordinates, we can adjust the angle:
- Adding 360°: (9, 75° + 360°) = (9, 435°)
- Subtracting 360°: (9, 75° – 360°) = (9, -285°)
Moreover, the radius can also be negative, which gives us another set of points:
- Using a negative radius: (-9, 75° + 180°) = (-9, 255°)
- Adding 360° to the angle: (-9, 255° + 360°) = (-9, 615°)
- Subtracting 360° from the angle: (-9, 255° – 360°) = (-9, -105°)
In conclusion, all polar coordinates of the given point are:
- (9, 75°)
- (9, 435°)
- (9, -285°)
- (-9, 255°)
- (-9, 615°)
- (-9, -105°)