Polar Coordinates Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard
Find all polar representations of the point with rectangular coordinates (0,โˆ’5)(0, -5) where โˆ’2ฯ€โ‰คฮธ<2ฯ€-2\pi \le \theta < 2\pi.

Solution

  1. 1
    r=5r = 5. The point is on the negative yy-axis, so ฮธ=3ฯ€2\theta = \frac{3\pi}{2} or equivalently ฮธ=โˆ’ฯ€2\theta = -\frac{\pi}{2}.
  2. 2
    We can also use r=โˆ’5r = -5 with ฮธ=ฯ€2\theta = \frac{\pi}{2} or ฮธ=โˆ’3ฯ€2\theta = -\frac{3\pi}{2}. So the representations are (5,3ฯ€2)(5, \frac{3\pi}{2}), (5,โˆ’ฯ€2)(5, -\frac{\pi}{2}), (โˆ’5,ฯ€2)(-5, \frac{\pi}{2}), (โˆ’5,โˆ’3ฯ€2)(-5, -\frac{3\pi}{2}).

Answer

(5,3ฯ€2),(5,โˆ’ฯ€2),(โˆ’5,ฯ€2),(โˆ’5,โˆ’3ฯ€2)\left(5, \frac{3\pi}{2}\right), \left(5, -\frac{\pi}{2}\right), \left(-5, \frac{\pi}{2}\right), \left(-5, -\frac{3\pi}{2}\right)
Unlike rectangular coordinates, polar representations are not unique. Adding 2ฯ€2\pi to ฮธ\theta gives the same point, and negating rr while adding ฯ€\pi to ฮธ\theta also gives the same point. This non-uniqueness is important when finding intersections of polar curves.

About Polar Coordinates

A coordinate system where each point in the plane is described by a distance rr from the origin and an angle ฮธ\theta from the positive xx-axis, written as (r,ฮธ)(r, \theta).

Learn more about Polar Coordinates โ†’

More Polar Coordinates Examples

Example 1 easy

Convert the polar coordinates [formula] to rectangular (Cartesian) coordinates.

Example 2 medium

Convert the rectangular point [formula] to polar coordinates with [formula] and [formula].

Example 3 medium

Convert the equation [formula] to polar form.

โ† All Polar Coordinates Examples Polar Coordinates Concept