Final answer:
The closest positive coterminal angle to 3π / 2 radians is 7π / 2 radians, and the closest negative coterminal angle is -π / 2 radians by subtracting or adding 2π radians.
Step-by-step explanation:
To find the closest positive angle and the closest negative angle that is coterminal to ⅓π / 2 radians, one must add or subtract multiples of 2π radians (360°), as a full rotation around the unit circle is 2π radians. Since 3π / 2 radians is the same as 270°, adding one full rotation gives us the closest positive coterminal angle: 3π / 2 + 2π = 7π / 2 radians. To find the closest negative coterminal angle, we subtract one full rotation yielding: 3π / 2 - 2π = -π / 2 radians.
The given angle is 3π / 2 radians. To find the closest positive angle, we add 2π to the given angle: 3π / 2 + 2π = 7π / 2. Therefore, the closest positive angle that is coterminal to 3π / 2 radians is 7π / 2 radians.
Similarly, to find the closest negative angle, we subtract 2π from the given angle: 3π / 2 - 2π = -π / 2. Therefore, the closest negative angle that is coterminal to 3π / 2 radians is -π / 2 radians.