The value of `sin^(-1)(cos""(3pi)/(5))` is

Options

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

Answer

To find the value of \(\sin^{-1}(\cos(\frac{3\pi}{5}))\), we can follow these steps:

1. **Use the complementary angle identity for cosine and sine:**
\[ \cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right) \]
Here, \(\theta = \frac{3\pi}{5}\).

2. **Rewrite the expression using the identity:**
\[ \sin^{-1}\left(\cos\left(\frac{3\pi}{5}\right)\right) = \sin^{-1}\left(\sin\left(\frac{\pi}{2} - \frac{3\pi}{5}\right)\right) \]

3. **Simplify the argument of the sine function:**
\[ \frac{\pi}{2} - \frac{3\pi}{5} = \frac{5\pi}{10} - \frac{6\pi}{10} = -\frac{\pi}{10} \]

4. **Substitute back into the expression:**
\[ \sin^{-1}\left(\sin\left(-\frac{\pi}{10}\right)\right) \]

5. **Use the property of the inverse sine function:**
\[ \sin^{-1}(\sin(\theta)) = \theta \quad \text{if} \quad -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \]
Since \(-\frac{\pi}{10}\) is within the range \(-\frac{\pi}{2} \leq -\frac{\pi}{10} \leq \frac{\pi}{2}\), we can directly write:
\[ \sin^{-1}\left(\sin\left(-\frac{\pi}{10}\right)\right) = -\frac{\pi}{10} \]

Therefore, the value of \(\sin^{-1}(\cos(\frac{3\pi}{5}))\) is \(-\frac{\pi}{10}\).