How do you find all the polar coordinates of P=(-1, -2pi/3)? I've read that you're supposed to add 2pi to find the coterminals but when I do that I get different answers then in the book. The book answer is (1, 2pi/3+ (2n+ 1) pi) and (-1, -2pi/3+ 2npi). How in the world do you get those answers?

you are correct to say that by adding 2π , one rotation , we point ourselves in the same direction,

how about 4π ? , or 6π or n(2π) where n is whole number?
so now the 2nd of their answer should make sense they have simply added multiples of 2π to get
(-1, -2π/3) = (-1, -2π/3 +2nπ)

now look at their 1st answer.
Did you notice that the -1 changed to +1 ?
That would have been a reversal of direction of 180° or π
let n = 0
we get (-1, -2π/3) ---> (+1, 2π/3 + π) or (1, 5π/3)
let's look at this in degrees .....
(-1, -120°) ---> (1, 120 + 180) = (1,300)

At this point we should realize that their 1st answer cannot be correct, since we don't end up at the same endpoint.
(-1, -120°) means: point yourself in the direction of -120
but then go -1 unit, or go 1 unit in the OPPOSITE direction, so the simplest version of
(-1,-120°) = (1,60°) or (1, π/3)

To show it is wrong .....
There has to be an integer value of n such that
2π/3 + (2n+1)π = π/3
2/3 + 2n+1 = 1/3
2n = 4/3
n = 2/3, but n has to be an integer.
so they are wrong.

To find all the polar coordinates of a point, you need to understand the concept of coterminal angles. Coterminal angles are angles that have the same terminal side but differ in their initial side. Adding or subtracting a multiple of 2π to an angle will give you its coterminal angles.

In this case, you have a point P with Cartesian coordinates of (-1, -2π/3). To find the polar coordinates, you need to convert these Cartesian coordinates to polar form (r, θ), where r represents the distance from the origin to the point, and θ represents the angle from the positive x-axis to the line segment connecting the origin and the point.

First, let's find the value of r. The distance between the origin and P can be found using the distance formula:

r = sqrt((-1)^2 + (-2π/3)^2) = sqrt(1 + 4π^2/9)

Now, let's find the value of θ. To determine the angle, you can use the atan2 function (or inverse tangent) by taking the y-coordinate and x-coordinate (or -2π/3 and -1, respectively) of the point P:

θ = atan2(-2π/3, -1)

Now, you have the initial polar coordinate (r, θ) for the point P. However, you also need to find all the coterminal angles.

To find the coterminal angles, you add or subtract a multiple of 2π to the initial angle θ. In this case, the book provides two sets of coterminal angles:

(1, 2π/3 + (2n + 1)π) and (-1, -2π/3 + 2nπ)

For the first set (1, 2π/3 + (2n + 1)π), notice that (2n + 1) represents an odd integer. This means that as you change the value of n, you will get different coterminal angles. Adding π to 2π/3 ensures that the angles are not repeated.

For the second set (-1, -2π/3 + 2nπ), here, 2nπ represents any multiple of 2π. By adding or subtracting multiples of 2π, you are finding the coterminal angles that lie on the negative x-axis.

By using these formulas, you can find all the polar coordinates (r, θ) for the given Cartesian coordinates (-1, -2π/3), as well as their corresponding coterminal angles.