Final answer:
To match the pairs of coordinates representing the same point, we must identify equivalent angles in the polar coordinate system and consider the radius' sign. After adjusting angles by full rotations and changing the sign of the radius, we find the matching pairs.
Step-by-step explanation:
The student's question involves matching pairs of coordinates that represent the same point in a polar coordinate system. To find the matching pairs, we need to consider that a polar coordinate consists of a radius and an angle, where the radius can be positive or negative and the angle can be represented in multiple ways by adding or subtracting full rotations (2π). Here, we are looking for coordinates that, despite being represented differently, denote the same point.
- (3, 5π/4) and (-3, -3π/4) are one pair as flipping the radius sign and adding π (180 degrees) to the angle gives us the same point.
- (-3, 3π/4) and (3, -5π/4) are also a match when we consider the same flip of the radius sign and adjusting the angle.
- (-3, 5π/2) and (3, -3π/2) represent the same point since 5π/2 is equivalent to -3π/2 after subtracting 2π, and flipping the radius changes the direction to match.
- (-3, 3π/2) and (3, π/2) represent the same point after flipping the radius sign and recognizing that -3π/2 is equivalent to π/2 when considering the angle as moving in the opposite direction.
For the remaining pairs, the principle followed is similar: adjusting the angle by multiples of 2π and changing the sign of the radius will yield matching pairs for the coordinates as listed.