the next arrow case for which the terminal point has the same coordinates with the terminal point for a2 is…

Answer:

Question 1:

1. For the first - part (coordinates of the terminal point): Without seeing the full - context of the graph and the value of (t=\frac{\pi}{2}), assuming a standard unit - circle or sine - wave context, if (t = \frac{\pi}{2}), the coordinates of the terminal point ((x,y)) are ((0,1)).
2. The value of the (\sin(t)) function at (t=\frac{\pi}{2}) is (1).

Question 2:

In a periodic function (such as the sine function), the period of (y = \sin(t)) is (2\pi). Points with the same coordinates repeat every period. If (A2) is a point on the sine - wave, the next point with the same coordinates is (A4) (assuming a standard labeling of points on a periodic graph). So the answer is (A4).

Question 3:

Since the period of the sine function (y=\sin(t)) is (2\pi), if a point has the same coordinates as another point, the difference in the (t) - values is a multiple of the period. The value of (t) corresponding to the point with the same coordinates as (A2) (the next such point) is (2\pi). So the answer is (2\pi).

Explanation:

Step 1: Recall sine - function properties

The sine function (y = \sin(t)) has a period of (2\pi), and on the unit - circle, for (t=\frac{\pi}{2}), (x = 0) and (y = 1), and (\sin(t)=y).

Step 2: Identify periodicity of points

Points on a periodic function (like sine) with the same coordinates repeat every period.

Step 3: Determine (t) value for repeated point

Since the period is (2\pi), the (t) value for the next point with the same coordinates as a given point is (2\pi) more than the original (t) value.