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The expression given is \( \sec(120^\circ) \).

The secant function is defined as the reciprocal of the cosine function:

\[
\sec(\theta) = \frac{1}{\cos(\theta)}
\]

Now, let's calculate \( \sec(120^\circ) \):

1. The angle \(120^\circ\) lies in the second quadrant, where the cosine function is negative.
2. The reference angle for \(120^\circ\) is \(180^\circ - 120^\circ = 60^\circ\).
3. We know that \( \cos(60^\circ) = \frac{1}{2} \).
4. Since cosine is negative in the second quadrant, \( \cos(120^\circ) = -\frac{1}{2} \).
5. Therefore, \( \sec(120^\circ) = \frac{1}{\cos(120^\circ)} = \frac{1}{-\frac{1}{2}} = -2 \).

Thus, the exact value of \( \sec(120^\circ) \) is:

\[
\sec(120^\circ) = -2
\]

Would you like a detailed explanation of how the reference angle and sign of cosine are determined?

Here are 5 related questions to explore further:
1. What is the relationship between secant and cosine for angles in different quadrants?
2. How do you find the exact values of other trigonometric functions at \(120^\circ\)?
3. How would you calculate \( \sec(240^\circ) \)?
4. What are the key steps to determine the reference angle for any angle?
5. How does the unit circle help in calculating secant for different angles?

**Tip:** Always check which quadrant an angle is in before determining the sign of the trigonometric functions.

Find the exact value of the expression: sec(120°).

Learn how to find the exact value of sec(120°) using trigonometric identities and reference angles. This problem explains key concepts, including the secant and cosine functions and how to use them for angles in different quadrants. Suitable for students in Grades 9-11.

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