Answer:
The all polar coordinates of P are:
(3 , -π/3) , (3 , 5π/3) , (-3 , 2π/3) , (-3 , -4π/3)
Step-by-step explanation:
* Lets study the polar coordinates of a point
- In polar coordinates there is an infinite number of coordinates
for a given point.
- The polar coordinates of a point (x , y) is (r , θ), where
r = √ ( x2 + y2 )
θ = tan-1 ( y / x )
# Ex: the following four points are all coordinates for the same point.
* (5 , π/3) = (5 , −5π/3) = (−5 , 4π/3) =(−5 , −2π/3)
- These four points only represent the coordinates of the point without
rotating more than once
- So the point (r,θ) can be represented by any of the following
coordinate pairs (r , θ + 2π n) and (−r , θ + (2n + 1) π), where n is
any integer.
* Now lets solve the problem
∵ P = (3 , -π/3)
∵ (r , θ + 2πn)
∴ r = 3 an d Ф = -π/3
- let n = 1
∴ P = (3 , -π/3 + 2π)
∴ P = (3 , 5π/3)
∵ P =(3 , -π/3)
∵ P = (-r , θ + (2n + 1) π)
Let n = 0
∴ P = (-3 , -π/3 + (2×0 + 1) π)
∴ P = (-3 , -π/3 + (0 + 1) π)
∴ P = (-3 , -π/3 + π)
∴ P = (-3 , 2π/3)
∵ P =(3 , -π/3)
∵ P = (-r , θ + (2n + 1) π)
Let n = -1
∴ P = (-3 , -π/3 + (2(-1) + 1) π)
∴ P = (-3 , -π/3 + (-2 + 1) π)
∴ P = (-3 , -π/3 + -π) = (-3 , -4π/3)
∴ P = (-3 , -4π/3)
