d) \( \frac{1}{2} \) *\#4.) Determine which of the following angles are not coterminal to \( \frac{\pi}{2} \). Select all that apply. a) \( -\frac{\pi}{2} \) b) \( -\frac{3 \pi}{2} \) c) \( \frac{7 \pi}{2} \) d) \( \frac{5 \pi}{2} \)

To determine which angles are not coterminal with \( \frac{\pi}{2} \), we need to understand the concept of coterminal angles. Two angles are coterminal if they differ by a multiple of \( 2\pi \). This means that for an angle \( \theta \), any angle of the form \( \theta + 2k\pi \) (where \( k \) is an integer) is coterminal with \( \theta \). Given the angle \( \frac{\pi}{2} \), we can express the condition for coterminality as follows: \[ \text{Angle} = \frac{\pi}{2} + 2k\pi \] Now, let's analyze each option: 1. **Option a: \( -\frac{\pi}{2} \)** - To check if it is coterminal, we can set up the equation: \[ -\frac{\pi}{2} = \frac{\pi}{2} + 2k\pi \] Rearranging gives: \[ -\pi = 2k\pi \implies k = -\frac{1}{2} \] Since \( k \) is not an integer, \( -\frac{\pi}{2} \) is **not coterminal**. 2. **Option b: \( -\frac{3\pi}{2} \)** - Check for coterminality: \[ -\frac{3\pi}{2} = \frac{\pi}{2} + 2k\pi \] Rearranging gives: \[ -2\pi = 2k\pi \implies k = -1 \] Since \( k \) is an integer, \( -\frac{3\pi}{2} \) is **coterminal**. 3. **Option c: \( \frac{7\pi}{2} \)** - Check for coterminality: \[ \frac{7\pi}{2} = \frac{\pi}{2} + 2k\pi \] Rearranging gives: \[ 3\pi = 2k\pi \implies k = \frac{3}{2} \] Since \( k \) is not an integer, \( \frac{7\pi}{2} \) is **not coterminal**. 4. **Option d: \( \frac{5\pi}{2} \)** - Check for coterminality: \[ \frac{5\pi}{2} = \frac{\pi}{2} + 2k\pi \] Rearranging gives: \[ 2\pi = 2k\pi \implies k = 1 \] Since \( k \) is an integer, \( \frac{5\pi}{2} \) is **coterminal**. ### Conclusion The angles that are **not coterminal** to \( \frac{\pi}{2} \) are: - a) \( -\frac{\pi}{2} \) - c) \( \frac{7\pi}{2} \)