To find all polar coordinates of the point P given in the form of the ordered pair (1, π/3), we need to understand the concept of polar coordinates.
Polar coordinates are represented as (r, θ), where:
- r: The distance from the origin to the point.
- θ: The angle measured from the positive x-axis to the line connecting the origin to the point.
In this case, the point P is specified as:
- r = 1: This means the distance from the origin (0, 0) to P is 1 unit.
- θ = π/3: This angle corresponds to 60 degrees in the Cartesian coordinate system.
Now, the beauty of polar coordinates is that multiple representations exist for the same point. Specifically, the polar coordinates can be rotated by adding or subtracting full circles (2π radians) and also by flipping the radius
Let’s derive the other polar coordinates for point P:
- By adding full rotations:
– (1, π/3 + 2πk) for any integer k.
– This gives: (1, π/3), (1, 7π/3), (1, 13π/3), etc. - By subtracting full rotations:
– Similarly: (1, π/3 – 2πk) for any integer k.
– This results in: (1, -5π/3), (1, -11π/3), etc. - By flipping the radius:
– We can also represent the point using the negative value of the radius: (-1, π/3 + π) = (-1, 4π/3).
– Therefore, other coordinates can be: (-1, 4π/3), (-1, 10π/3), (-1, 16π/3), etc.
In conclusion, the complete set of polar coordinates for the point P(1, π/3) is:
- (1, π/3 + 2πk)
- (1, π/3 – 2πk)
- (-1, π/3 + π + 2πk)
Here, k is any integer. This flexibility and multiplicity of representations highlight the elegance of polar coordinates!