`sin(3theta+alpha)+sin(3theta-alpha)+sin(alpha-theta)-sin(alpha+theta)=cosalpha`

Answer

To solve the equation \( \sin(3\theta + \alpha) + \sin(3\theta - \alpha) + \sin(\alpha - \theta) - \sin(\alpha + \theta) = \cos \alpha \), we can follow these steps:

### Step 1: Combine the sine terms
Using the sine addition and subtraction formulas:
\[ \sin(a + b) + \sin(a - b) = 2 \sin a \cos b \]
we can rewrite the first two terms:
\[ \sin(3\theta + \alpha) + \sin(3\theta - \alpha) = 2 \sin(3\theta) \cos(\alpha) \]
Now, for the last two terms:
\[ \sin(\alpha - \theta) - \sin(\alpha + \theta) = -2 \sin(\theta) \cos(\alpha) \]
Thus, the equation becomes:
\[ 2 \sin(3\theta) \cos(\alpha) - 2 \sin(\theta) \cos(\alpha) = \cos(\alpha) \]

### Step 2: Factor out common terms
We can factor out \( 2 \cos(\alpha) \) from the left side:
\[ 2 \cos(\alpha) (\sin(3\theta) - \sin(\theta)) = \cos(\alpha) \]

### Step 3: Divide both sides by \( \cos(\alpha) \)
Assuming \( \cos(\alpha) \neq 0 \), we can divide both sides by \( \cos(\alpha) \):
\[ 2 (\sin(3\theta) - \sin(\theta)) = 1 \]

### Step 4: Simplify the equation
This simplifies to:
\[ \sin(3\theta) - \sin(\theta) = \frac{1}{2} \]

### Step 5: Use the sine triple angle formula
Using the identity \( \sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta) \), we can substitute:
\[ 3\sin(\theta) - 4\sin^3(\theta) - \sin(\theta) = \frac{1}{2} \]
This simplifies to:
\[ 2\sin(\theta) - 4\sin^3(\theta) = \frac{1}{2} \]

### Step 6: Rearranging the equation
Rearranging gives us:
\[ 4\sin^3(\theta) - 2\sin(\theta) + \frac{1}{2} = 0 \]
Multiplying through by 2 to eliminate the fraction:
\[ 8\sin^3(\theta) - 4\sin(\theta) + 1 = 0 \]

### Step 7: Let \( x = \sin(\theta) \)
Let \( x = \sin(\theta) \), we have:
\[ 8x^3 - 4x + 1 = 0 \]

### Step 8: Solve the cubic equation
Now we can solve the cubic equation \( 8x^3 - 4x + 1 = 0 \). We can use numerical methods or factorization if possible.

### Step 9: Finding the roots
Using the Rational Root Theorem or synthetic division, we can find the roots. Let's assume we find one root \( x = \frac{1}{2} \).

### Step 10: Finding \( \theta \)
If \( \sin(\theta) = \frac{1}{2} \), then:
\[ \theta = \frac{\pi}{6} + n\pi \quad \text{or} \quad \theta = \frac{5\pi}{6} + n\pi \]
for \( n \in \mathbb{Z} \).

### Final Solution
Thus, the general solutions for \( \theta \) are:
\[ \theta = \frac{\pi}{6} + n\pi \quad \text{or} \quad \theta = \frac{5\pi}{6} + n\pi \]