Find the value of ` sin^(-1) ( cos. ( 33pi)/5)` .

Options

A. `-pi/5`
B. `-pi/10`
C. `pi/5`
D. `pi/10`

Answer

To find the value of \( \sin^{-1} \left( \cos \left( \frac{33\pi}{5} \right) \right) \), we can follow these steps:

### Step 1: Simplify the angle \( \frac{33\pi}{5} \)
First, we simplify \( \frac{33\pi}{5} \) by finding an equivalent angle within the range of \( [0, 2\pi) \).

\[ \frac{33\pi}{5} = 6\pi + \frac{3\pi}{5} \]

Since \( 6\pi \) is a multiple of \( 2\pi \), we can subtract \( 6\pi \) to find the equivalent angle:

\[ \frac{33\pi}{5} - 6\pi = \frac{33\pi}{5} - \frac{30\pi}{5} = \frac{3\pi}{5} \]

### Step 2: Use the cosine identity
Now we can rewrite the cosine function:

\[ \cos\left(\frac{33\pi}{5}\right) = \cos\left(\frac{3\pi}{5}\right) \]

### Step 3: Find \( \sin^{-1} \left( \cos\left(\frac{3\pi}{5}\right) \right) \)
Next, we need to evaluate \( \sin^{-1} \left( \cos\left(\frac{3\pi}{5}\right) \right) \).

Using the identity \( \cos\theta = \sin\left(\frac{\pi}{2} - \theta\right) \):

\[ \cos\left(\frac{3\pi}{5}\right) = \sin\left(\frac{\pi}{2} - \frac{3\pi}{5}\right) \]

### Step 4: Calculate \( \frac{\pi}{2} - \frac{3\pi}{5} \)
Now we calculate:

\[ \frac{\pi}{2} - \frac{3\pi}{5} = \frac{5\pi}{10} - \frac{6\pi}{10} = -\frac{\pi}{10} \]

### Step 5: Substitute back into the sine inverse
Now we can substitute this back into our expression:

\[ \sin^{-1} \left( \cos\left(\frac{3\pi}{5}\right) \right) = \sin^{-1} \left( \sin\left(-\frac{\pi}{10}\right) \right) \]

### Step 6: Evaluate \( \sin^{-1} \left( \sin\left(-\frac{\pi}{10}\right) \right) \)
Since \( -\frac{\pi}{10} \) is within the range of \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), we have:

\[ \sin^{-1} \left( \sin\left(-\frac{\pi}{10}\right) \right) = -\frac{\pi}{10} \]

### Final Answer
Thus, the value of \( \sin^{-1} \left( \cos \left( \frac{33\pi}{5} \right) \right) \) is:

\[ \boxed{-\frac{\pi}{10}} \]