1. Which of the following are possible pains of polar coordinates for the point with rectangular coordinates $(0,2)$ ? a. $\left(2, \frac{\pi}{2}\right)$ b. $\left(2, \frac{7 \pi}{2}\right)$ d. $\left(-2, \frac{7 \pi}{2}\right)$ e. $\left(-2,-\frac{\pi}{2}\right)$ f. $\left(2,-\frac{\pi}{2}\right)$

Step 1 ：The rectangular coordinates (0,2) represent a point on the y-axis, 2 units above the origin. In polar coordinates, this point can be represented as (r, θ) where r is the distance from the origin and θ is the angle measured counter-clockwise from the positive x-axis.

Step 2 ：The distance from the origin to the point (0,2) is 2 units, so r = 2. The angle θ is 90 degrees or π/2 radians because the point is directly above the origin on the y-axis.

Step 3 ：We can add any multiple of 2π to θ and it will still represent the same point. So, θ could also be 5π/2, 9π/2, etc.

Step 4 ：If we take r to be negative, then we must add π to the angle to get the same point. So, another valid polar coordinate for the point (0,2) could be (-2, 3π/2), (-2, 7π/2), etc.

Step 5 ：The polar coordinates (2, 7π/2), (-2, 7π/2), and (2, -π/2) all map to the point (0,2) in rectangular coordinates. However, the polar coordinate (-2, -π/2) maps to the point (0,-2), not (0,2).

Step 6 ：Final Answer: The possible pairs of polar coordinates for the point with rectangular coordinates (0,2) are \(\boxed{\left(2, \frac{\pi}{2}\right), \left(2, \frac{7 \pi}{2}\right), \left(-2, \frac{7 \pi}{2}\right)}\)