Question

For the given point `A(-4, frac{pi}{4})`, how do you list three different pairs of polar coordinates that represent this point such that `-2pi \le \theta \le 2pi`?

Answer 1

This question is designed to demonstrate your understanding of polar coordinates. The restriction to a range of angles is a good help in making you think about positive and negative radius values.

For polar coordinates, the two coordinates represent a radius and an angle. From some point which is defined as the center, all points on a plane can be described by knowing the distance from the center and the rotation angle from an axis defined as the zero angle.

The first thing to recognize in this problem is that the radius -4 will describe points on a circle a distance of 4 from the center. So this coordinate could be written as either 4 or -4. But to get to the same coordinate, the angle will have to be in the other direction. A full rotation is `2pi`. So half a rotation will be `pi`. We must add or subtract `pi` from the angle if we change -4 to 4.

`(4,pi/4 + pi)`
`(4,pi/4 - pi)`

These are both within the angle limitations stated in the problem:
`-2pi <= theta <= 2pi`

The other possibility to to keep the direction the same, but rotate the angle by a complete turn by adding or subtracting `2pi`.

`(-4,pi/4 + 2pi)`
`(-4,pi/4 - 2pi)`

Since the first of these answers give an angle larger than `2pi` as the question required, only the second should be given.