To solve \(\tan\left(-\frac{\pi}{2}\right)\), let's break down the process:

1. **Identify the angle:**
   \[
   -\frac{\pi}{2} \text{ radians}
   \]

2. **Understand the angle position:**
   The angle \(-\frac{\pi}{2}\) corresponds to a clockwise rotation of \(\frac{\pi}{2}\) radians (or 90 degrees) from the positive x-axis.

3. **Evaluate the tangent at this angle:**
   The tangent function is undefined at \(\theta = \pm \frac{\pi}{2}\) because the tangent of an angle is the ratio of the sine to the cosine, and \(\cos\left(\pm\frac{\pi}{2}\right) = 0\). This makes the tangent function undefined since division by zero is not allowed.

So, the solution is:
\[
\tan\left(-\frac{\pi}{2}\right) \text{ is undefined.}
\]

Would you like more details on any step, or do you have any questions?

Here are 5 related questions you might find interesting:
1. Why is the tangent function undefined at \(\theta = \frac{\pi}{2}\)?
2. What is \(\sin\left(-\frac{\pi}{2}\right)\)?
3. How do you graph the tangent function over one period?
4. What are the values of \(\tan(\theta)\) at key angles like \(0\), \(\frac{\pi}{4}\), and \(\frac{\pi}{2}\)?
5. How does the tangent function behave as \(\theta\) approaches \(\frac{\pi}{2}\)?

**Tip:** When solving trigonometric functions, remember that \(\tan(\theta)\) is undefined where \(\cos(\theta) = 0\).

Explore the solution to \(\tan\left(-\frac{\pi}{2}\right)\), which demonstrates the behavior of the tangent function at key angles. Understand why the tangent function is undefined at \(\theta = \pm \frac{\pi}{2}\) due to division by zero. Suitable for advanced high school students and beyond.

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