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The question in the image asks for the exact value of \( \sec^{-1}(2) \).

To solve this:

The secant function, \( \sec(\theta) \), is the reciprocal of the cosine function, meaning:
\[
\sec(\theta) = \frac{1}{\cos(\theta)}
\]

Given \( \sec(\theta) = 2 \), we have:
\[
\frac{1}{\cos(\theta)} = 2
\]

Solving for \( \cos(\theta) \), we get:
\[
\cos(\theta) = \frac{1}{2}
\]

We need to find the angle \( \theta \) such that \( \cos(\theta) = \frac{1}{2} \). This occurs at \( \theta = \frac{\pi}{3} \) radians, or 60° in degrees.

Thus, the exact value of \( \sec^{-1}(2) \) is \( \frac{\pi}{3} \).

Would you like more details on this process or have any questions?

Here are 5 related questions for further exploration:
1. What is the relationship between secant and cosine in trigonometry?
2. How do you find the inverse of trigonometric functions like secant and cosine?
3. Why is the cosine of \( \frac{\pi}{3} \) equal to \( \frac{1}{2} \)?
4. What is the domain and range of the secant function?
5. How do you evaluate other inverse trigonometric functions such as \( \sec^{-1}(3) \)?

**Tip**: For trigonometric functions, always check for standard angles in radians and degrees to simplify calculations.

Find the exact value of sec^{-1}(2) using trigonometric principles. This problem explores the relationship between secant and cosine, solving for θ where sec(θ) = 2. Ideal for students in Grades 10-12, this solution involves standard angle analysis and the properties of inverse trigonometric functions.

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