Value of `sec^(-1)(sec.(4pi)/3)` is

Options

A. `pi/3`
B. `(2pi)/3`
C. `(4pi)/3`
D. `pi/3`

Answer

To find the value of \( \sec^{-1}(\sec(4\pi/3)) \), we will follow these steps:

### Step 1: Understand the function
The function \( \sec^{-1}(x) \) gives the angle whose secant is \( x \). The principal value of \( \sec^{-1}(x) \) lies in the range \( [0, \pi/2) \cup (\pi/2, \pi] \).

### Step 2: Set up the equation
Let \( y = \sec^{-1}(\sec(4\pi/3)) \). This implies:
\[ \sec(y) = \sec(4\pi/3) \]

### Step 3: Evaluate \( \sec(4\pi/3) \)
To find \( \sec(4\pi/3) \), we first need to find \( \cos(4\pi/3) \):
\[ \cos(4\pi/3) = \cos\left(\pi + \frac{\pi}{3}\right) = -\cos\left(\frac{\pi}{3}\right) = -\frac{1}{2} \]
Thus,
\[ \sec(4\pi/3) = \frac{1}{\cos(4\pi/3)} = \frac{1}{-\frac{1}{2}} = -2 \]

### Step 4: Solve for \( y \)
Now we have:
\[ \sec(y) = -2 \]
This means:
\[ y = \sec^{-1}(-2) \]

### Step 5: Find the angle corresponding to \( \sec^{-1}(-2) \)
The angle \( y \) must be in the range \( [0, \pi/2) \cup (\pi/2, \pi] \). The secant function is negative in the second quadrant. The angle whose secant is \( -2 \) is:
\[ y = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \]

### Step 6: Conclusion
Thus, the value of \( \sec^{-1}(\sec(4\pi/3)) \) is:
\[ \sec^{-1}(\sec(4\pi/3)) = \frac{2\pi}{3} \]

### Final Answer:
\[ \sec^{-1}(\sec(4\pi/3)) = \frac{2\pi}{3} \]

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