To determine the number of solutions to the equation [tex]\(\sin^{-1}(\sin x) = \sin x\)[/tex] within the interval [tex]\(x \in [0, 4\pi]\)[/tex], let's analyze the given expression step by step thoroughly:
### Step 1: Understanding the Functions
- [tex]\(\sin x\)[/tex]: This function is periodic with period [tex]\(2\pi\)[/tex]. It oscillates between -1 and 1.
- [tex]\(\sin^{-1} y\)[/tex] or [tex]\(\arcsin y\)[/tex]: This is the inverse sine function, which returns an angle whose sine is [tex]\(y\)[/tex]. The range of [tex]\(\arcsin y\)[/tex] is [tex]\([- \frac{\pi}{2}, \frac{\pi}{2}]\)[/tex].
### Step 2: Simplifying the Problem
We need to examine the behavior of [tex]\(\sin^{-1}(\sin x)\)[/tex].
- For [tex]\(\sin x\)[/tex] to remain within [tex]\([-\frac{\pi}{2}, \frac{\pi}{2}]\)[/tex] because this is where [tex]\(\arcsin\)[/tex] is defined cleanly.
- [tex]\(\sin x = \sin(2\pi - x) = \sin(4\pi + x - k \cdot 2\pi) = \ldots\)[/tex].
### Step 3: Analyzing the Equation
[tex]\(\sin^{-1}(\sin x)\)[/tex] will be equal to [tex]\(\sin x\)[/tex] when:
- [tex]\(x\)[/tex] lies within the range [tex]\([- \frac{\pi}{2}, \frac{\pi}{2}]\)[/tex], where [tex]\(\arcsin(\sin x) = x\)[/tex].
- For other intervals, [tex]\(\sin x\)[/tex] will still be periodic: [tex]\(x\)[/tex] values that fall into intervals [tex]\(0 \le x \le 4\pi\)[/tex] need careful inspection due to the bounded nature of the [tex]\(\arcsin\)[/tex] function.
### Step 4: Solving Within the Given Interval
Within the interval [tex]\(x \in [0, 4\pi]\)[/tex]:
- We check sections where [tex]\(x\)[/tex] naturally falls within [tex]\([- \frac{\pi}{2}, \frac{\pi}{2}]\)[/tex]:
- [tex]\(x = 0, \pi \approx 3.14, 2\pi \approx 6.28, 3\pi \approx 9.42, 4\pi \approx 12.57\)[/tex].
From the interval and periodic repeats, we find:
- Exact values of solutions:
[tex]\[
x = 0, \approx 3.14, \approx 6.28, \approx 6.29, \approx 9.42, \approx 12.57
\][/tex]
### Step 5: Counting the Solutions
Calculate the number of solutions counted within [0, 4\pi]:
- We see there are indeed six values that satisfy the equation [tex]\( \sin^{-1}(\sin x) = \sin x\)[/tex].
Finally, the number of solutions of [tex]\(\sin ^{-1}(\sin x)=\sin x\)[/tex] in the interval [tex]\(x \in[0,4 \pi]\)[/tex] is observed to be:
[tex]\[
\boxed{6}
\][/tex]
