Solve `sin^(-1) (sin 6x) = x, x in [0,pi]`

Explanation

To solve the equation \( \sin^{-1}(\sin 6x) = x \) for \( x \) in the interval \([0, \pi]\), we can follow these steps:

### Step 1: Understand the function
The function \( \sin^{-1}(y) \) gives the angle whose sine is \( y \). The range of \( \sin^{-1}(y) \) is \([- \frac{\pi}{2}, \frac{\pi}{2}]\). Therefore, for the equation \( \sin^{-1}(\sin 6x) = x \) to hold, \( x \) must also lie within this range.

### Step 2: Set up the equation
We start with the equation:
\[
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