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The problem asks to find the exact values of trigonometric functions using the unit circle for \( s = \frac{\pi}{2} \). We are tasked with evaluating \( \sin \left( -\frac{\pi}{2} \right) \), \( \cos \left( -\frac{\pi}{2} \right) \), and \( \tan \left( \frac{\pi}{2} \right) \).

Let’s break down each one step by step:

1. **For \( \sin \left( -\frac{\pi}{2} \right) \)**:
   - In the unit circle, \( -\frac{\pi}{2} \) is equivalent to rotating clockwise by 90 degrees. This puts us at the point \( (0, -1) \).
   - Since \( \sin \theta \) is the \( y \)-coordinate on the unit circle, \( \sin \left( -\frac{\pi}{2} \right) = -1 \).

2. **For \( \cos \left( -\frac{\pi}{2} \right) \)**:
   - The same point \( (0, -1) \) applies, and the \( \cos \theta \) is the \( x \)-coordinate.
   - Thus, \( \cos \left( -\frac{\pi}{2} \right) = 0 \).

3. **For \( \tan \left( \frac{\pi}{2} \right) \)**:
   - The tangent function is defined as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
   - At \( \frac{\pi}{2} \), \( \sin \left( \frac{\pi}{2} \right) = 1 \) and \( \cos \left( \frac{\pi}{2} \right) = 0 \).
   - Since division by zero is undefined, \( \tan \left( \frac{\pi}{2} \right) \) is **undefined**.

### Now for the multiple-choice questions:
- **First question (select the correct sine value)**:
   \( \sin \left( -\frac{\pi}{2} \right) = -1 \), so the answer is **A**.

- **Second question (select the correct cosine value)**:
   \( \cos \left( -\frac{\pi}{2} \right) = 0 \), so the answer is **A**.

- **Third question (tan \( \frac{\pi}{2} \))**:
   \( \tan \left( \frac{\pi}{2} \right) \) is undefined, so the answer is **B**.

Would you like further details or clarifications? Here are five related questions to consider:

1. How do you find trigonometric values for angles beyond \( 0 \) and \( \pi \)?
2. Why is the tangent function undefined at certain angles?
3. Can trigonometric functions have values greater than 1 or less than -1?
4. What is the significance of the unit circle in trigonometry?
5. How does symmetry in the unit circle help simplify trigonometric calculations?

**Tip:** Always remember that cosine gives the x-coordinate and sine gives the y-coordinate of an angle on the unit circle.

Use the unit circle, along with the definitions of the circular functions, to find the exact values for the given functions when s = -π/2: sin(-π/2), cos(-π/2), and tan(π/2).

Learn how to calculate the exact values for sin(-π/2), cos(-π/2), and tan(π/2) using the unit circle. This example explores key trigonometric concepts such as sine, cosine, and tangent for special angles. Ideal for high school students in Grades 10-12.

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