What is the value of the following functions (using principal value)
`sin^(-1)("sin"(4pi)/(5))`

Answer

To find the value of the expression \( \sin^{-1}(\sin(4\pi/5)) \), we will follow these steps:

### Step 1: Evaluate \( \sin(4\pi/5) \)

We can express \( 4\pi/5 \) in a more manageable form:
\[ 4\pi/5 = \pi - \pi/5 \]
This means we can use the sine subtraction formula.

### Step 2: Use the sine identity

Using the sine identity \( \sin(\pi - x) = \sin(x) \), we have:
\[ \sin(4\pi/5) = \sin(\pi - \pi/5) = \sin(\pi/5) \]

### Step 3: Substitute back into the original expression

Now we substitute this back into our original expression:
\[ \sin^{-1}(\sin(4\pi/5)) = \sin^{-1}(\sin(\pi/5)) \]

### Step 4: Apply the property of inverse sine

The property of the inverse sine function states that:
\[ \sin^{-1}(\sin(x)) = x \quad \text{for } x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]
However, since \( \pi/5 \) is not in the range of \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), we need to find the principal value.

### Step 5: Determine the principal value

Since \( 4\pi/5 \) is in the second quadrant, where sine is positive, we can conclude:
\[ \sin^{-1}(\sin(4\pi/5)) = \pi - 4\pi/5 = \pi/5 \]

### Final Answer

Thus, the value of \( \sin^{-1}(\sin(4\pi/5)) \) is:
\[ \frac{\pi}{5} \]

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