Final answer:
The terminal point P(x, y) on the unit circle for the value t = 9π/2 is the same as for π/2, which is (0, 1), because 9π/2 radians corresponds to a full circle (8π/2) plus an additional π/2 radians. The radius of the unit circle is always 1, hence the distance from the origin to P remains constant.
Step-by-step explanation:
To find the terminal point P(x, y) on the unit circle determined by a given value of t, where t is the angle in radians, we note that the unit circle completes a full revolution every 2π radians. The question asks for the terminal point when t = 9π/2. To determine this point, we need to recognize that the angle given will correspond to an angle within the first revolution of 2π radians.
Since 9π/2 radians is the same as 4π + π/2 radians, and we know that every full revolution (every multiple of 2π) brings us back to the starting point on the unit circle, we can ignore the complete revolutions and only concentrate on the π/2 radians which is a quarter of a revolution. The point on the unit circle at π/2 is (0, 1), which means that the terminal point P(x, y) after an angle of 9π/2 is also (0, 1), where x = 0 and y = 1.
Note that the unit circle's radius is invariant under rotations which means the distance from the origin to any point on the circle always equals 1, thereby confirming that a rotation by 9π/2 radians does not change this distance. This concept can be mathematically expressed as x² + y² = 1.