The principal values of `cos^(-1)(-sin(7pi)/(6))` is

Options

A. `(5pi)/(3)`
B. `(7pi)/(6)`
C. `(pi)/(3)`
D. none of these

Explanation

To find the principal value of \( \cos^{-1}\left(-\sin\left(\frac{7\pi}{6}\right)\right) \), we will follow these steps:

### Step 1: Simplify \( \sin\left(\frac{7\pi}{6}\right) \)
We start by simplifying \( \sin\left(\frac{7\pi}{6}\right) \). We can express \( \frac{7\pi}{6} \) as \( \pi + \frac{\pi}{6} \).

Using the sine addition formula, we know:
\[ \sin(\pi + x) = -\sin(x) \]
Thus,
\[ \sin\left(\frac{7\pi}{6}\right) = \sin\left(\pi + \frac{\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right) \]

### Step 2: Find \( \sin\left(\frac{\pi}{6}\right) \)
Now, we know:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]
So,
\[ \sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2} \]

### Step 3: Substitute back into the expression
Now we substitute this back into our original expression:
\[ \cos^{-1}\left(-\sin\left(\frac{7\pi}{6}\right)\right) = \cos^{-1}\left(-\left(-\frac{1}{2}\right)\right) = \cos^{-1}\left(\frac{1}{2}\right) \]

### Step 4: Find \( \cos^{-1}\left(\frac{1}{2}\right) \)
The principal value of \( \cos^{-1}\left(\frac{1}{2}\right) \) is:
\[ \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \]

### Final Answer
Thus, the principal value of \( \cos^{-1}\left(-\sin\left(\frac{7\pi}{6}\right)\right) \) is:
\[ \frac{\pi}{3} \]

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