Answer:
The other two other representation are (6 , -4π/3) and (-6 , -π/3)
Explanation:
* Lets revise some important facts about the polar form of a point
- In polar coordinates there is an infinite number of coordinates for a
given point
- The point (r , θ) can be represented by any of the following coordinate
pairs (r , θ+2πn) , (-r , θ + [2n+1]π) , where n is any integer
* Now lets solve the problem
∵ A point has polar coordinates (6 , 2π/3)
- We can find many points as the same with this point
- The point (r , θ) can be represented by any of the following coordinate
pairs(r , θ + 2πn) and (-r , θ + (2n + 1)π), where n is any integer.
∵ The angle in [-2π , 2π]
∵ r = 6 and Ф = 2π/3
- Let n = -1
∴ (r , Ф + 2πn) = (6 , 2π/3 + 2π(-1)) = (6 , 2π/3 - 2π) = (6 , -4π/3)
* One point is (6 , -4π/3)
∴ (-r , θ + (2n + 1)π) = (-6 , 2π/3 + (2(-1) + 1)π) = (-6 , 2π/3 + (-2 + 1)π)
∴ (-r , θ + (2n + 1)π) = (-6 , 2π/3 + (-1)π) = (-6 , 2π/3 - π)
∴ (-r , θ + (2n + 1)π) = (-6 , -π/3)
* One point is (-6 , -π/3)