Solve the system:
`x sin a + y sin 2a + z sin 3a = sin 4a`
`x sin b + y sin 2b + z sin 3b = sin 4b`
`x sin c + y sin 2c + z sin 3c = sin 4c`

Answer

To solve the system of equations given by:

1. \( x \sin a + y \sin 2a + z \sin 3a = \sin 4a \)
2. \( x \sin b + y \sin 2b + z \sin 3b = \sin 4b \)
3. \( x \sin c + y \sin 2c + z \sin 3c = \sin 4c \)

we will follow these steps:

### Step 1: Rewrite the equations using sine double angle and triple angle formulas
We know that:
- \( \sin 2a = 2 \sin a \cos a \)
- \( \sin 3a = 3 \sin a - 4 \sin^3 a \)
- \( \sin 4a = 2 \sin 2a \cos 2a = 4 \sin a \cos a \cos 2a \)

Substituting these into the first equation gives:
\[ x \sin a + y (2 \sin a \cos a) + z (3 \sin a - 4 \sin^3 a) = 4 \sin a \cos a \cos 2a \]
Factoring out \( \sin a \) (assuming \( \sin a \neq 0 \)):
\[ x + 2y \cos a + z (3 - 4 \sin^2 a) = 4 \cos a \cos 2a \]

### Step 2: Simplify the equation
Using the identity \( \cos 2a = 2 \cos^2 a - 1 \), we can rewrite \( 4 \cos a \cos 2a \):
\[ 4 \cos a (2 \cos^2 a - 1) = 8 \cos^3 a - 4 \cos a \]
Thus, the equation becomes:
\[ x + 2y \cos a + z (3 - 4 \sin^2 a) = 8 \cos^3 a - 4 \cos a \]

### Step 3: Substitute \( \sin^2 a \) with \( 1 - \cos^2 a \)
Substituting \( \sin^2 a = 1 - \cos^2 a \):
\[ x + 2y \cos a + z (3 - 4(1 - \cos^2 a)) = 8 \cos^3 a - 4 \cos a \]
This simplifies to:
\[ x + 2y \cos a + z (4 \cos^2 a - 1) = 8 \cos^3 a - 4 \cos a \]

### Step 4: Rearranging the equation
Rearranging gives us a polynomial in \( \cos a \):
\[ 8 \cos^3 a - (4 + 4z) \cos^2 a + (2y + 4) \cos a + (x + z) = 0 \]

### Step 5: Repeat for other equations
Repeat the same process for equations 2 and 3:
- For \( b \), we will have a similar polynomial in \( \cos b \).
- For \( c \), we will have a similar polynomial in \( \cos c \).

### Step 6: Roots of the polynomial
The values \( \cos a \), \( \cos b \), and \( \cos c \) will be the roots of the respective polynomials derived from the equations.

### Conclusion
The solution to the system of equations is the values of \( x, y, z \) that satisfy the derived polynomial equations for \( \cos a, \cos b, \cos c \).