`sec^(-1)(-2)`

Explanation

To find the value of \( \sec^{-1}(-2) \), we can follow these steps:

### Step 1: Understand the Definition
The function \( \sec^{-1}(x) \) is the inverse of the secant function. It gives us the angle \( \theta \) such that \( \sec(\theta) = x \).

### Step 2: Set Up the Equation
We need to solve the equation:
\[ \sec(\theta) = -2 \]

### Step 3: Recall the Range of the Inverse Secant Function
The principal branch of the inverse secant function, \( \sec^{-1}(x) \), has a range of:
\[ \theta \in \left[0, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \pi\right] \]
This means that the angle \( \theta \) must be in the first or second quadrant.

### Step 4: Find the Reference Angle
Since the secant function is negative, we are looking for an angle in the second quadrant. The reference angle for \( \sec(\theta) = 2 \) is:
\[ \theta = \frac{\pi}{3} \]
because \( \sec\left(\frac{\pi}{3}\right) = 2 \).

### Step 5: Determine the Angle in the Second Quadrant
To find the angle in the second quadrant corresponding to this reference angle, we use:
\[ \theta = \pi - \frac{\pi}{3} \]
Calculating this gives:
\[ \theta = \frac{3\pi}{3} - \frac{\pi}{3} = \frac{2\pi}{3} \]

### Step 6: Conclusion
Thus, the value of \( \sec^{-1}(-2) \) is:
\[ \sec^{-1}(-2) = \frac{2\pi}{3} \]