If sin3x + Asin2x + Bsinx = Csin^3x, ∀ x ∈ R, then the value of |A + B + C| is 

The given equation is:

\[
\sin 3x + A\sin 2x + B\sin x = C\sin^3 x, \quad \forall x \in \mathbb{R}.
\]

We need to determine the value of \( |A + B + C| \).

### Step 1: Expand the trigonometric identities.

First, recall the trigonometric identities for \( \sin 3x \) and \( \sin 2x \):

\[
\sin 3x = 3\sin x - 4\sin^3 x,
\]
\[
\sin 2x = 2\sin x \cos x = 2\sin x \sqrt{1 - \sin^2 x}.
\]

Given that \( \sin 3x + A\sin 2x + B\sin x = C\sin^3 x \) holds for all \( x \), we need to match coefficients of similar terms on both sides of the equation.

### Step 2: Substitute \( t = \sin x \).

Let \( t = \sin x \). Then the equation becomes:

\[
3t - 4t^3 + A(2t \sqrt{1 - t^2}) + Bt = Ct^3.
\]

We need this equation to hold for all values of \( t \in [-1, 1] \). To simplify the analysis, consider specific values of \( t \).

### Step 3: Substitute specific values for \( t \) to find \( A \), \( B \), and \( C \).

1. **For \( t = 0 \):**

\[
3(0) - 4(0)^3 + A(2 \cdot 0 \cdot \sqrt{1 - 0^2}) + B(0) = C(0)^3.
\]

This simplifies to:

\[
0 = 0,
\]

which is true but doesn't provide any information about \( A \), \( B \), or \( C \).

2. **For \( t = 1 \):**

\[
3(1) - 4(1)^3 + A(2 \cdot 1 \cdot \sqrt{1 - 1^2}) + B(1) = C(1)^3.
\]

This simplifies to:

\[
3 - 4 + B = C.
\]

So, we have:

\[
B - 1 = C. \quad \text{(Equation 1)}
\]

3. **For \( t = -1 \):**

\[
3(-1) - 4(-1)^3 + A(2 \cdot -1 \cdot \sqrt{1 - (-1)^2}) + B(-1) = C(-1)^3.
\]

This simplifies to:

\[
-3 - 4 - B = -C.
\]

So, we have:

\[
-7 - B = -C \implies C = B + 7. \quad \text{(Equation 2)}
\]

### Step 4: Solve the system of equations.

From Equation 1: \( C = B - 1 \).

From Equation 2: \( C = B + 7 \).

Equating these:

\[
B - 1 = B + 7.
\]

This equation is inconsistent, indicating a mistake. Let’s re-examine the approach.

### Step 5: Reevaluate the coefficients.

Instead of focusing on specific values of \( t \), expand the expression:

Given:

\[
\sin 3x + A\sin 2 = txtmlemkingFILLbootstrapb### Please

Explore how to solve the trigonometric equation sin3x + Asin2x + Bsinx = Csin^3x for constants A, B, and C. This detailed analysis covers trigonometric identities and their applications in solving equations. Suitable for advanced high school students and beyond.

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