Find the principal values of the following :
`sec^(-1) (-2)`

Answer

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

### Step 1: Understand the definition of the inverse secant function
The function \( \sec^{-1}(x) \) gives us an angle \( \theta \) such that \( \sec(\theta) = x \). The range of \( \sec^{-1}(x) \) is \( [0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi] \).

### Step 2: Set up the equation
We need to find \( \theta \) such that:
\[ \sec(\theta) = -2 \]

### Step 3: Convert secant to cosine
Recall that \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Therefore, we can rewrite the equation as:
\[ \frac{1}{\cos(\theta)} = -2 \]
This implies:
\[ \cos(\theta) = -\frac{1}{2} \]

### Step 4: Determine the angles where cosine is negative
The cosine function is negative in the second and third quadrants. The angles corresponding to \( \cos(\theta) = -\frac{1}{2} \) are:
\[ \theta = \frac{2\pi}{3} \quad \text{and} \quad \theta = \frac{4\pi}{3} \]

### Step 5: Identify the principal value
Since we are looking for the principal value of \( \sec^{-1}(-2) \), we must select the angle that lies within the range of \( \sec^{-1}(x) \), which is \( [0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi] \). The angle \( \frac{2\pi}{3} \) falls within the range \( (\frac{\pi}{2}, \pi] \).

Thus, the principal value of \( \sec^{-1}(-2) \) is:
\[ \theta = \frac{2\pi}{3} \]

### Final Answer
The principal value of \( \sec^{-1}(-2) \) is:
\[ \frac{2\pi}{3} \]

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