To solve \(\text{arcsec}(2)\), we need to recall the definition and properties of the secant and inverse secant (arcsecant) functions.

### Definition of arcsecant:
The arcsecant function, \(\text{arcsec}(x)\), is the inverse of the secant function. This means that if:
\[
\theta = \text{arcsec}(x)
\]
then:
\[
\sec(\theta) = x
\]
and \(\theta\) is typically restricted to the range \(0 \leq \theta < \pi\), with \(\theta \neq \frac{\pi}{2}\).

### Solve \(\text{arcsec}(2)\):
We are solving for \(\theta\) such that:
\[
\sec(\theta) = 2
\]
The secant is the reciprocal of the cosine:
\[
\sec(\theta) = \frac{1}{\cos(\theta)}
\]
So we can rewrite the equation as:
\[
\frac{1}{\cos(\theta)} = 2
\]
Now, solving for \(\cos(\theta)\):
\[
\cos(\theta) = \frac{1}{2}
\]

### Step-by-Step Solution:
We need to find \(\theta\) such that \(\cos(\theta) = \frac{1}{2}\), and \(\theta\) lies in the valid range for \(\text{arcsec}(x)\).

- The cosine of \(\theta = \frac{\pi}{3}\) is \(\frac{1}{2}\).
- The secant function is positive in the first and fourth quadrants, but arcsec is defined in the first and second quadrants.

Thus, the correct value for \(\theta\) is:
\[
\theta = \frac{\pi}{3}
\]

Therefore:
\[
\text{arcsec}(2) = \frac{\pi}{3}
\]

Would you like any further details on this, or a step-by-step breakdown?

### Follow-up Questions:
1. What is the domain and range of the arcsecant function?
2. How does the secant function relate to the cosine function?
3. Can we solve \(\text{arcsec}(-2)\)? How?
4. What are some other important inverse trigonometric functions?
5. How do we solve for \(\text{arccos}(x)\) and \(\text{arcsin}(x)\)?

**Tip:** Always check the domain restrictions when working with inverse trigonometric functions to avoid errors.

Learn how to solve arcsec(2) with a detailed explanation of inverse trigonometric functions and their properties. This solution includes step-by-step guidance on using the secant and cosine relationships to find θ = π/3. Ideal for high school students studying trigonometry.

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