The value of `sin^(-1)(cos(33pi)/5)` is
(a)`(3pi)/5`
(b) `-pi/(10)`
(c) `pi/(10)`
(d) `(7pi)/5`

Options

A. `(3pi)/(5)`
B. `(-7pi)/(5)`
C. `(pi)/(10)`
D. `(-pi)/(10)`

Explanation

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

### Step 1: Simplify \( 33\pi/5 \)
We can express \( 33\pi/5 \) in a more manageable form. We can separate it into two parts:
\[ 33\pi/5 = \frac{30\pi}{5} + \frac{3\pi}{5} = 6\pi + \frac{3\pi}{5} \]

**Hint:** Break down the angle into a multiple of \( 2\pi \) plus a smaller angle to simplify the calculation.

### Step 2: Determine the cosine value
Since \( 6\pi \) represents three complete revolutions (as \( 6\pi = 3 \times 2\pi \)), the angle \( 6\pi + \frac{3\pi}{5} \) lies in the first quadrant. Therefore, we can find:
\[ \cos(33\pi/5) = \cos(6\pi + \frac{3\pi}{5}) = \cos(\frac{3\pi}{5}) \]

**Hint:** Remember that cosine is periodic with a period of \( 2\pi \).

### Step 3: Rewrite \( \frac{3\pi}{5} \)
Next, we can express \( \frac{3\pi}{5} \) in terms of a sine function:
\[ \frac{3\pi}{5} = \pi - \frac{2\pi}{5} \]
Thus, we have:
\[ \cos\left(\frac{3\pi}{5}\right) = -\cos\left(\frac{2\pi}{5}\right) \]

**Hint:** Use the identity \( \cos(\pi - x) = -\cos(x) \) to relate angles.

### Step 4: Find \( \sin^{-1}(\cos(33\pi/5)) \)
Now, we can find:
\[ \sin^{-1}(\cos(33\pi/5)) = \sin^{-1}(-\cos(\frac{2\pi}{5})) \]
Using the identity \( \sin^{-1}(-x) = -\sin^{-1}(x) \), we get:
\[ \sin^{-1}(-\cos(\frac{2\pi}{5})) = -\sin^{-1}(\cos(\frac{2\pi}{5})) \]

**Hint:** Remember that \( \sin^{-1}(-x) = -\sin^{-1}(x) \).

### Step 5: Evaluate \( \sin^{-1}(\cos(\frac{2\pi}{5})) \)
We can express \( \cos(\frac{2\pi}{5}) \) as \( \sin(\frac{\pi}{2} - \frac{2\pi}{5}) = \sin(\frac{\pi}{2} - \frac{2\pi}{5}) \). Therefore:
\[ \sin^{-1}(\cos(\frac{2\pi}{5})) = \frac{\pi}{2} - \frac{2\pi}{5} \]
Calculating this gives:
\[ \frac{\pi}{2} - \frac{2\pi}{5} = \frac{5\pi}{10} - \frac{4\pi}{10} = \frac{\pi}{10} \]

**Hint:** Use the complementary angle identity to find the sine inverse.

### Step 6: Final Result
Putting it all together, we have:
\[ \sin^{-1}(\cos(33\pi/5)) = -\left(\frac{\pi}{10}\right) = -\frac{\pi}{10} \]

Thus, the final answer is:
\[ \boxed{-\frac{\pi}{10}} \]