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.