To solve the equation \((3 + \cos x)^2 = 4 - 2\sin^8 x\) in the interval \([0, 9\pi)\), we will follow these steps:
### Step 1: Analyze the left-hand side
The left-hand side of the equation is \((3 + \cos x)^2\). The cosine function, \(\cos x\), ranges from \(-1\) to \(1\). Therefore, we can find the range of \(3 + \cos x\):
\[
3 + \cos x \text{ ranges from } 3 - 1 = 2 \text{ to } 3 + 1 = 4.
\]
Thus, the range of \((3 + \cos x)^2\) is:
\[
(2)^2 = 4 \text{ to } (4)^2 = 16.
\]
### Step 2: Analyze the right-hand side
The right-hand side of the equation is \(4 - 2\sin^8 x\). The sine function, \(\sin x\), also ranges from \(-1\) to \(1\), which means \(\sin^8 x\) ranges from \(0\) to \(1\) (since it is raised to an even power). Therefore, we can find the range of \(4 - 2\sin^8 x\):
\[
\sin^8 x \text{ ranges from } 0 \text{ to } 1 \Rightarrow 4 - 2\sin^8 x \text{ ranges from } 4 - 2(0) = 4 \text{ to } 4 - 2(1) = 2.
\]
Thus, the range of \(4 - 2\sin^8 x\) is:
\[
2 \text{ to } 4.
\]
### Step 3: Set the ranges equal
Now we have the ranges:
- Left-hand side: \(4 \leq (3 + \cos x)^2 \leq 16\)
- Right-hand side: \(2 \leq 4 - 2\sin^8 x \leq 4\)
The two ranges overlap at the value \(4\). Therefore, we can set up the equation:
\[
(3 + \cos x)^2 = 4.
\]
### Step 4: Solve for \(\cos x\)
Taking the square root of both sides, we have:
\[
3 + \cos x = 2 \quad \text{or} \quad 3 + \cos x = -2.
\]
1. For \(3 + \cos x = 2\):
\[
\cos x = 2 - 3 = -1.
\]
2. For \(3 + \cos x = -2\):
\[
\cos x = -2 - 3 = -5 \quad \text{(not possible since } \cos x \text{ must be between } -1 \text{ and } 1).
\]
### Step 5: Find solutions for \(\cos x = -1\)
The equation \(\cos x = -1\) holds true at:
\[
x = (2n + 1)\pi \quad \text{for integers } n.
\]
### Step 6: Determine valid \(n\) values in the interval \([0, 9\pi)\)
We need to find values of \(n\) such that:
\[
0 \leq (2n + 1)\pi < 9\pi.
\]
Dividing through by \(\pi\):
\[
0 \leq 2n + 1 < 9.
\]
This simplifies to:
\[
-1 < 2n < 8 \Rightarrow 0 \leq n < 4.
\]
Thus, \(n\) can take values \(0, 1, 2, 3\).
### Step 7: List the solutions
Calculating the corresponding \(x\) values:
- For \(n = 0\): \(x = 1\pi = \pi\)
- For \(n = 1\): \(x = 3\pi\)
- For \(n = 2\): \(x = 5\pi\)
- For \(n = 3\): \(x = 7\pi\)
### Conclusion
The number of solutions to the equation \((3 + \cos x)^2 = 4 - 2\sin^8 x\) in the interval \([0, 9\pi)\) is \(4\).
### Final Answer
The number of solutions is \(4\).
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