`-(2pi)/(5)` is the principal value of

Options

A. `"cos"^(-1)("cos""(7pi)/(5))`
B. `"sin"^(-1)("sin""(7pi)/(5))`
C. `sec^(-1)(sec"(7pi)/(5))`
D. None of these

Answer

To find out which inverse trigonometric function has a principal value of \(-\frac{2\pi}{5}\), we can follow these steps:

### Step 1: Rewrite the Principal Value
We start with the principal value given in the question:
\[ -\frac{2\pi}{5} \]

### Step 2: Relate to Sine Function
We can express this value in terms of the sine function. We know that:
\[ -\frac{2\pi}{5} = -\sin^{-1}(\sin(\frac{2\pi}{5})) \]
This is because the sine inverse function gives us the angle whose sine is the given value.

### Step 3: Use the Property of Sine
Using the property of sine, we can rewrite the negative angle:
\[ -\sin^{-1}(\sin(\frac{2\pi}{5})) = \sin^{-1}(-\sin(\frac{2\pi}{5})) \]
Now, we can use the identity that relates sine of angles:
\[ \sin(\pi + \theta) = -\sin(\theta) \]
Thus, we can express \(-\sin(\frac{2\pi}{5})\) as:
\[ -\sin(\frac{2\pi}{5}) = \sin(\pi + \frac{2\pi}{5}) = \sin(\frac{7\pi}{5}) \]

### Step 4: Substitute Back
Now we can substitute back into our expression:
\[ -\frac{2\pi}{5} = \sin^{-1}(\sin(\frac{7\pi}{5})) \]

### Conclusion
Thus, we conclude that:
\[ -\frac{2\pi}{5} \text{ is the principal value of } \sin^{-1}(\sin(\frac{7\pi}{5})) \]

### Final Answer
The correct option is:
\[ \sin^{-1}(\sin(\frac{7\pi}{5})) \]

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