The value of \( \sec \left( \frac{7\pi}{4} \right) \) can be calculated as follows:

1. First, recognize that \( \sec(\theta) = \frac{1}{\cos(\theta)} \). So we need to find \( \cos \left( \frac{7\pi}{4} \right) \).
2. The angle \( \frac{7\pi}{4} \) is located in the fourth quadrant, as it is equivalent to \( 2\pi - \frac{\pi}{4} \), or \( 360^\circ - 45^\circ = 315^\circ \).
3. In the fourth quadrant, the cosine of the angle is positive, and \( \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \).

Therefore, \( \cos \left( \frac{7\pi}{4} \right) = \frac{\sqrt{2}}{2} \).

4. Now, calculate \( \sec \left( \frac{7\pi}{4} \right) \):

\[
\sec \left( \frac{7\pi}{4} \right) = \frac{1}{\cos \left( \frac{7\pi}{4} \right)} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}
\]

Thus, \( \sec \left( \frac{7\pi}{4} \right) = \sqrt{2} \).

Would you like more details on trigonometric identities?

Here are some related questions:
1. What is the value of \( \sec \left( \frac{5\pi}{4} \right) \)?
2. Can you calculate \( \csc \left( \frac{7\pi}{4} \right) \)?
3. What are the values of sine and tangent at \( \frac{7\pi}{4} \)?
4. How do trigonometric functions behave in each quadrant?
5. What are the exact values of trigonometric functions at special angles?

**Tip**: Always remember the signs of trigonometric functions in each quadrant: cosine is positive in the fourth quadrant.

Learn how to calculate the secant of 7π/4. This problem involves core trigonometric concepts such as sec(θ) = 1 / cos(θ) and determining the cosine in different quadrants. Suitable for high school students in Grades 10-12.

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