To evaluate \( \sec\left(-\frac{11\pi}{4}\right) \), let's follow these steps:

### Step 1: Simplify the angle
Since the secant function is periodic with a period of \( 2\pi \), we reduce the given angle to an equivalent angle within the interval \( [0, 2\pi] \). 

\[
-\frac{11\pi}{4} + 2\pi = -\frac{11\pi}{4} + \frac{8\pi}{4} = -\frac{3\pi}{4}
\]

Now, we are evaluating \( \sec\left(-\frac{3\pi}{4}\right) \).

### Step 2: Use symmetry properties
The secant function is an even function, meaning that \( \sec(-x) = \sec(x) \). Therefore,

\[
\sec\left(-\frac{3\pi}{4}\right) = \sec\left(\frac{3\pi}{4}\right)
\]

### Step 3: Evaluate \( \sec\left(\frac{3\pi}{4}\right) \)
We know that \( \frac{3\pi}{4} \) is in the second quadrant, where the cosine function is negative. The reference angle is \( \frac{\pi}{4} \), and the cosine of \( \frac{\pi}{4} \) is \( \frac{1}{\sqrt{2}} \).

Since \( \cos\left(\frac{3\pi}{4}\right) = -\frac{1}{\sqrt{2}} \), the secant is the reciprocal of cosine:

\[
\sec\left(\frac{3\pi}{4}\right) = \frac{1}{\cos\left(\frac{3\pi}{4}\right)} = \frac{1}{-\frac{1}{\sqrt{2}}} = -\sqrt{2}
\]

### Final Answer:
\[
\sec\left(-\frac{11\pi}{4}\right) = -\sqrt{2}
\]

Let me know if you'd like further clarification!

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Here are five related questions:
1. What is the value of \( \sec(\frac{\pi}{4}) \)?
2. How do we calculate \( \csc\left(-\frac{5\pi}{3}\right) \)?
3. What is the period of the secant function?
4. How does the secant function behave in different quadrants?
5. Can you find \( \sec(2\pi + \frac{\pi}{3}) \)?

**Tip:** Always check if the angle can be simplified by using periodicity of trigonometric functions to make calculations easier.

Learn how to evaluate sec(-11π/4) by simplifying the angle using trigonometric periodicity and symmetry properties. This step-by-step guide covers secant function calculations, including reducing angles and applying sec(x) = 1/cos(x). Suitable for Grades 10-12 students.

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