Check out the polar coordinate pairs that label the same point. Several pairs may be possible.

A) (2, 2pi/3)
B) (2, -pi/3)
C) (-3, pi)
D) (2, 7pi/3)
E) (3,0)
F) (-3,0)
G) (-2, -pi/3)
H) -3, 2pi)

AG

CE
FH

To find out which polar coordinate pairs label the same point, we need to convert the polar coordinates to Cartesian coordinates and compare the points.

A) (2, 2π/3):
To convert to Cartesian coordinates:
x = r * cos(θ) = 2 * cos(2π/3) = -1
y = r * sin(θ) = 2 * sin(2π/3) = √3

B) (2, -π/3):
To convert to Cartesian coordinates:
x = r * cos(θ) = 2 * cos(-π/3) = 1
y = r * sin(θ) = 2 * sin(-π/3) = -√3

C) (-3, π):
To convert to Cartesian coordinates:
x = r * cos(θ) = -3 * cos(π) = -3
y = r * sin(θ) = -3 * sin(π) = 0

D) (2, 7π/3):
To convert to Cartesian coordinates:
x = r * cos(θ) = 2 * cos(7π/3) = 1
y = r * sin(θ) = 2 * sin(7π/3) = -√3

E) (3, 0):
To convert to Cartesian coordinates:
x = r * cos(θ) = 3 * cos(0) = 3
y = r * sin(θ) = 3 * sin(0) = 0

F) (-3, 0):
To convert to Cartesian coordinates:
x = r * cos(θ) = -3 * cos(0) = -3
y = r * sin(θ) = -3 * sin(0) = 0

G) (-2, -π/3):
To convert to Cartesian coordinates:
x = r * cos(θ) = -2 * cos(-π/3) = -1
y = r * sin(θ) = -2 * sin(-π/3) = √3

H) (-3, 2π):
To convert to Cartesian coordinates:
x = r * cos(θ) = -3 * cos(2π) = -3
y = r * sin(θ) = -3 * sin(2π) = 0

Now we compare the Cartesian coordinates to see which points are the same:

A) (-1, √3)
B) (1, -√3)
C) (-3, 0)
D) (1, -√3)
E) (3, 0)
F) (-3, 0)
G) (-1, √3)
H) (-3, 0)

From the coordinates, we can see that pairs C, E, and F label the same point, since they have the same coordinates.
So the polar coordinate pairs that label the same point are:
C) (-3, π)
E) (3, 0)
F) (-3, 0)

To check which polar coordinate pairs label the same point, we need to convert each pair to Cartesian coordinates (x, y) and compare them.

The conversion from polar coordinates to Cartesian coordinates is given by:
x = r * cos(theta)
y = r * sin(theta)

Let's go through each polar coordinate pair and convert them to Cartesian coordinates:

A) (2, 2π/3):
x = 2 * cos(2π/3) ≈ -1
y = 2 * sin(2π/3) ≈ √3
Cartesian coordinates: (-1, √3)

B) (2, -π/3):
x = 2 * cos(-π/3) ≈ 1
y = 2 * sin(-π/3) ≈ -√3
Cartesian coordinates: (1, -√3)

C) (-3, π):
x = -3 * cos(π) = -3
y = -3 * sin(π) = 0
Cartesian coordinates: (-3, 0)

D) (2, 7π/3):
x = 2 * cos(7π/3) ≈ -1
y = 2 * sin(7π/3) ≈ -√3
Cartesian coordinates: (-1, -√3)

E) (3, 0):
x = 3 * cos(0) = 3
y = 3 * sin(0) = 0
Cartesian coordinates: (3, 0)

F) (-3, 0):
x = -3 * cos(0) = -3
y = -3 * sin(0) = 0
Cartesian coordinates: (-3, 0)

G) (-2, -π/3):
x = -2 * cos(-π/3) ≈ 1
y = -2 * sin(-π/3) ≈ -√3
Cartesian coordinates: (1, -√3)

H) (-3, 2π):
x = -3 * cos(2π) = -3
y = -3 * sin(2π) = 0
Cartesian coordinates: (-3, 0)

Now, we can compare the Cartesian coordinates to see which coordinate pairs label the same point:

The pairs (C) (-3, π) and (F) (-3, 0) both have the Cartesian coordinates (-3, 0), so they label the same point.

Answer: The polar coordinate pairs that label the same point are (C) (-3, π) and (F) (-3, 0).