Principal value of `cos^(-1)[cos((4pi)/(3))]` is

Options

A. `-(2pi)/(3)`
B. `-(4pi)/(6)`
C. `(4pi)/(3)`
D. `(2pi)/(3)`

Answer

To find the principal value of \( \cos^{-1} \left[ \cos \left( \frac{4\pi}{3} \right) \right] \), we can follow these steps:

### Step 1: Identify the angle in terms of known values
We can express \( \frac{4\pi}{3} \) in a more recognizable form:
\[ \frac{4\pi}{3} = \pi + \frac{\pi}{3} \]

### Step 2: Use the cosine addition formula
Using the cosine addition formula, we know:
\[ \cos(\pi + \theta) = -\cos(\theta) \]
Thus, we can write:
\[ \cos\left(\frac{4\pi}{3}\right) = \cos\left(\pi + \frac{\pi}{3}\right) = -\cos\left(\frac{\pi}{3}\right) \]

### Step 3: Calculate \( \cos\left(\frac{\pi}{3}\right) \)
We know that:
\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]
Therefore:
\[ \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2} \]

### Step 4: Substitute back into the inverse cosine function
Now we need to find:
\[ \cos^{-1}\left(-\frac{1}{2}\right) \]

### Step 5: Determine the principal value
The principal value of \( \cos^{-1}(x) \) is defined in the range \( [0, \pi] \). The angle whose cosine is \( -\frac{1}{2} \) in this range is:
\[ \cos^{-1}\left(-\frac{1}{2}\right) = \frac{2\pi}{3} \]

### Final Answer
Thus, the principal value of \( \cos^{-1}\left[\cos\left(\frac{4\pi}{3}\right)\right] \) is:
\[ \frac{2\pi}{3} \]

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