annotations for full credit. **\#2.) Which angle is coterminal with \( \frac{3 \pi}{2} \) ? a) \( -\frac{3 \pi}{2} \) b) \( -\frac{\pi}{2} \) c) \( \frac{\pi}{2} \) d) \( \frac{5 \pi}{2} \) (

To find an angle that is coterminal with \( \frac{3 \pi}{2} \), we can use the property that coterminal angles differ by integer multiples of \( 2\pi \). This means we can add or subtract \( 2\pi \) (or \( 360^\circ \) in degrees) to the given angle. ### Step 1: Identify the given angle The given angle is: \[ \theta = \frac{3 \pi}{2} \] ### Step 2: Find coterminal angles To find coterminal angles, we can use the formula: \[ \theta + 2k\pi \] where \( k \) is any integer. ### Step 3: Calculate coterminal angles 1. For \( k = 1 \): \[ \frac{3 \pi}{2} + 2\pi = \frac{3 \pi}{2} + \frac{4 \pi}{2} = \frac{7 \pi}{2} \] 2. For \( k = -1 \): \[ \frac{3 \pi}{2} - 2\pi = \frac{3 \pi}{2} - \frac{4 \pi}{2} = -\frac{\pi}{2} \] ### Step 4: List the options Now, let's compare the calculated coterminal angles with the provided options: - a) \( -\frac{3 \pi}{2} \) - b) \( -\frac{\pi}{2} \) (this is coterminal) - c) \( \frac{\pi}{2} \) - d) \( \frac{5 \pi}{2} \) ### Conclusion The angle that is coterminal with \( \frac{3 \pi}{2} \) is: \[ \boxed{-\frac{\pi}{2}} \] Thus, the correct answer is option **b)** \( -\frac{\pi}{2} \).