Solve `|sin3x+sinx|+|sin3x-sinx|=sqrt(3), -pi/2 lt theta lt pi/2`

Answer

To solve the equation \( | \sin(3x) + \sin(x) | + | \sin(3x) - \sin(x) | = \sqrt{3} \) for \( -\frac{\pi}{2} < x < \frac{\pi}{2} \), we can follow these steps:

### Step 1: Analyze the expression
We start with the equation:
\[ | \sin(3x) + \sin(x) | + | \sin(3x) - \sin(x) | = \sqrt{3} \]
Recall that the absolute value function can be simplified depending on the signs of the expressions inside.

### Step 2: Use the sine addition formulas
Using the sine addition formulas, we can express \( \sin(3x) \) in terms of \( \sin(x) \):
\[ \sin(3x) = 3\sin(x) - 4\sin^3(x) \]
This will help us rewrite the equation in terms of \( \sin(x) \).

### Step 3: Substitute and simplify
Substituting \( \sin(3x) \) into the equation gives:
\[ | (3\sin(x) - 4\sin^3(x)) + \sin(x) | + | (3\sin(x) - 4\sin^3(x)) - \sin(x) | = \sqrt{3} \]
This simplifies to:
\[ | 4\sin(x) - 4\sin^3(x) | + | 2\sin(x) - 4\sin^3(x) | = \sqrt{3} \]
Factoring out common terms:
\[ 4 | \sin(x)(1 - \sin^2(x)) | + | 2\sin(x) - 4\sin^3(x) | = \sqrt{3} \]

### Step 4: Analyze cases based on the signs
Next, we need to analyze the cases based on the values of \( \sin(x) \):
1. **Case 1:** \( \sin(x) \geq 0 \)
2. **Case 2:** \( \sin(x) < 0 \)

### Step 5: Solve for \( \sin(x) \)
For \( \sin(x) \geq 0 \):
\[ 4\sin(x)(1 - \sin^2(x)) + 2\sin(x) - 4\sin^3(x) = \sqrt{3} \]
This simplifies to:
\[ 4\sin(x) - 4\sin^3(x) + 2\sin(x) - 4\sin^3(x) = \sqrt{3} \]
Combining terms gives:
\[ 6\sin(x) - 8\sin^3(x) = \sqrt{3} \]

For \( \sin(x) < 0 \), the absolute values will change signs, and we will have:
\[ -4\sin(x)(1 - \sin^2(x)) - (2\sin(x) - 4\sin^3(x)) = \sqrt{3} \]

### Step 6: Find specific values
From the analysis, we can test specific angles where \( \sin(x) \) takes values that are easy to compute:
- Test \( x = \frac{\pi}{3} \):
\[ \sin\left(3 \cdot \frac{\pi}{3}\right) = \sin(\pi) = 0 \]
\[ |0 + \frac{\sqrt{3}}{2}| + |0 - \frac{\sqrt{3}}{2}| = \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} = \sqrt{3} \]
Thus, \( x = \frac{\pi}{3} \) is a solution.

- Test \( x = -\frac{\pi}{3} \):
\[ \sin\left(3 \cdot -\frac{\pi}{3}\right) = \sin(-\pi) = 0 \]
\[ |0 - \left(-\frac{\sqrt{3}}{2}\right)| + |0 + \left(-\frac{\sqrt{3}}{2}\right)| = \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} = \sqrt{3} \]
Thus, \( x = -\frac{\pi}{3} \) is also a solution.

### Final Solutions
The solutions to the equation are:
\[ x = \frac{\pi}{3}, -\frac{\pi}{3} \]