Consider the function `f(x) = sin^(-1)x`, having principal value branch `[(pi)/(2), (3pi)/(2)] and g(x) = cos^(-1)x`, having principal value brach `[0, pi]`
The value of `f(sin 10)` is

Options

A. `10 - 3pi`
B. `10 -2 pi`
C. `10 -(5pi)/(2)`
D. `(7pi)/(2) - 10`

Explanation

To find the value of \( f(\sin 10) \) where \( f(x) = \sin^{-1}(x) \) with the principal value branch \([ \frac{\pi}{2}, \frac{3\pi}{2} ]\), we can follow these steps:

### Step 1: Understand the function
The function \( f(x) = \sin^{-1}(x) \) is the inverse sine function. The range of this function is typically \([- \frac{\pi}{2}, \frac{\pi}{2}]\), but in this case, we are considering the principal value branch \([ \frac{\pi}{2}, \frac{3\pi}{2} ]\).

### Step 2: Calculate \( \sin 10 \)
We need to find \( \sin(10) \). Since \( 10 \) is in degrees, we can convert it to radians:
\[ 10^\circ = \frac{10 \pi}{180} = \frac{\pi}{18} \]
Now, we can find \( \sin(10^\circ) \).

### Step 3: Find \( f(\sin 10) \)
Now we need to evaluate \( f(\sin(10^\circ)) \):
\[ f(\sin(10^\circ)) = \sin^{-1}(\sin(10^\circ)) \]
Since \( 10^\circ \) is within the range of the standard sine function, we have:
\[ f(\sin(10^\circ)) = 10^\circ = \frac{\pi}{18} \]

### Step 4: Adjust for the principal value branch
However, since we are considering the principal value branch of \( f(x) \) which is \([ \frac{\pi}{2}, \frac{3\pi}{2} ]\), we need to adjust our result.

To do this, we can use the property of the sine function:
\[ \sin(180^\circ - x) = \sin(x) \]
We can express \( 10^\circ \) in terms of \( 180^\circ \):
\[ \sin(10^\circ) = \sin(180^\circ - 10^\circ) = \sin(170^\circ) \]
Thus, we can write:
\[ f(\sin(10^\circ)) = \pi - \frac{\pi}{18} = \frac{18\pi}{18} - \frac{\pi}{18} = \frac{17\pi}{18} \]

### Final Answer
Therefore, the value of \( f(\sin(10)) \) is:
\[ \frac{17\pi}{18} \]