Find the value of `"sec" (13 (pi)/3)`

Answer

To find the value of \( \sec\left(\frac{13\pi}{3}\right) \), we will follow these steps:

### Step 1: Simplify the angle
First, we need to simplify \( \frac{13\pi}{3} \) to find an equivalent angle within the range of \( 0 \) to \( 2\pi \).

To do this, we can subtract \( 2\pi \) (which is equivalent to \( \frac{6\pi}{3} \)) from \( \frac{13\pi}{3} \):

\[ \frac{13\pi}{3} - 2\pi = \frac{13\pi}{3} - \frac{6\pi}{3} = \frac{7\pi}{3} \]

Now, \( \frac{7\pi}{3} \) is still greater than \( 2\pi \), so we subtract \( 2\pi \) again:

\[ \frac{7\pi}{3} - 2\pi = \frac{7\pi}{3} - \frac{6\pi}{3} = \frac{\pi}{3} \]

Thus, we have:

\[ \frac{13\pi}{3} \equiv \frac{\pi}{3} \quad (\text{mod } 2\pi) \]

### Step 2: Use the secant identity
We know that:

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

So, we can write:

\[ \sec\left(\frac{13\pi}{3}\right) = \sec\left(\frac{\pi}{3}\right) \]

### Step 3: Find the cosine value
Next, we need to find \( \cos\left(\frac{\pi}{3}\right) \). The value of \( \cos\left(\frac{\pi}{3}\right) \) is:

\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

### Step 4: Calculate the secant
Now, substituting the cosine value into the secant function:

\[ \sec\left(\frac{\pi}{3}\right) = \frac{1}{\cos\left(\frac{\pi}{3}\right)} = \frac{1}{\frac{1}{2}} = 2 \]

### Final Answer
Thus, the value of \( \sec\left(\frac{13\pi}{3}\right) \) is:

\[ \sec\left(\frac{13\pi}{3}\right) = 2 \]

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