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.