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) $$ \frac{5 \pi}{2} $$

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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) $$ \frac{7\pi}{2} $$
The angles that are not coterminal with $$ \frac{\pi}{2} $$ are: - a) $$ -\frac{\pi}{2} $$ - c) $$ \frac{7\pi}{2} $$

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) \( \frac{5 \pi}{2} \)

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