To find all the polar coordinates of a point with a given angle, we need to understand the relationship between polar and Cartesian coordinates. In polar coordinates, a point in the plane is represented as (r, θ), where r is the radial distance from the origin, and θ is the angle measured from the positive x-axis.
Given the angle θ = 9π/5, we first convert this angle into a more manageable form. The angle 9π/5 can be adjusted by subtracting 2π (which is equivalent to a full rotation) to get an equivalent angle within the range of 0 to 2π. This yields:
9π/5 – 2π = 9π/5 – 10π/5 = -π/5
This means that the angle θ = 9π/5 is equivalent to θ = -π/5.
Polar coordinates also allow for perspectives at different radial distances. For a point P, its polar coordinates can be expressed in an infinite number of ways by varying the radial distance. A general representation can be given as:
- (r, θ) for any r > 0 (where θ = 9π/5)
- (-r, θ + π) (this reflects the point across the origin) for any r > 0
Therefore, some examples of polar coordinates for point P where θ = 9π/5 include:
- (1, 9π/5)
- (2, 9π/5)
- (3, 9π/5)
- (-1, 9π/5 + π = 9π/5 + 5π/5 = 14π/5)
- (-2, 9π/5 + π = 14π/5)
- (-3, 9π/5 + π = 14π/5)
In conclusion, point P can be represented with an infinite set of polar coordinates defined by varying the value of r. The essential angles remain 9π/5 and 14π/5.