Final answer:
The polar coordinates that represent the same point as (4, -π/2) are c) (4, 3π/2), because an angle of 3π/2 counterclockwise is the same as -π/2 clockwise, placing the point on the negative y-axis.
Step-by-step explanation:
The question is asking which polar coordinates represent the same point as (4, -π/2). Polar coordinates specify a point by its distance from the origin, r, and the angle, θ, from the positive x-axis. The given point (4, -π/2) has a distance of 4 from the origin and is π/2 radians clockwise from the positive x-axis, placing it on the negative y-axis.
Option a) (4, π/2) represents a distance of 4 from the origin and π/2 radians counterclockwise from the positive x-axis, which would also place it on the negative y-axis, but it's the opposite direction from the given point.
Option b) (-4, π/2) represents a distance of 4 from the origin, but since the distance is negative, it is in the opposite direction of π/2, which would place the point on the positive y-axis.
Option c) (4, 3π/2) represents a distance of 4 from the origin and 3π/2 radians counterclockwise from the positive x-axis, which is directly equivalent to (4, -π/2). This is because an angle of 3π/2 counterclockwise is the same as -π/2 clockwise, both placing the point on the negative y-axis.
Option d) (-4, -π/2) has a distance of 4 from the origin with a negative sign, suggesting a direction opposite of -π/2, which places the point on the positive y-axis.
The correct answer is c) (4, 3π/2) as it represents the same point in polar coordinates as the given point (4, -π/2).