We are given the equation:

\[
A = \frac{\sin(x)}{\sin\left(\frac{\pi}{3} - x\right)}
\]

We need to solve for \(x\).

### Step-by-Step Solution

1. **Start by expressing the equation** in a more workable form:
   
\[
A = \frac{\sin(x)}{\sin\left(\frac{\pi}{3} - x\right)}
\]

2. **Apply the identity** for \(\sin\left(\frac{\pi}{3} - x\right)\). Using the sine subtraction identity:

\[
\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b)
\]

Substitute \(a = \frac{\pi}{3}\) and \(b = x\):

\[
\sin\left(\frac{\pi}{3} - x\right) = \sin\left(\frac{\pi}{3}\right)\cos(x) - \cos\left(\frac{\pi}{3}\right)\sin(x)
\]

Since \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\) and \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\), this becomes:

\[
\sin\left(\frac{\pi}{3} - x\right) = \frac{\sqrt{3}}{2} \cos(x) - \frac{1}{2} \sin(x)
\]

3. **Substitute this back into the equation**:

\[
A = \frac{\sin(x)}{\frac{\sqrt{3}}{2} \cos(x) - \frac{1}{2} \sin(x)}
\]

4. **Multiply both sides** by the denominator on the right-hand side:

\[
A\left(\frac{\sqrt{3}}{2} \cos(x) - \frac{1}{2} \sin(x)\right) = \sin(x)
\]

5. **Expand the left-hand side**:

\[
A\left(\frac{\sqrt{3}}{2} \cos(x)\right) - A\left(\frac{1}{2} \sin(x)\right) = \sin(x)
\]

6. **Group terms involving \(\sin(x)\)**:

\[
A\frac{\sqrt{3}}{2} \cos(x) = \sin(x) + A\frac{1}{2} \sin(x)
\]

7. **Factor out \(\sin(x)\) on the right-hand side**:

\[
A\frac{\sqrt{3}}{2} \cos(x) = \left(1 + \frac{A}{2}\right)\sin(x)
\]

8. **Solve for \(x\)**:

Now, depending on the value of \(A\), we can solve this trigonometric equation for \(x\). If a specific value for \(A\) is given, we can proceed with calculating \(x\).

---

Would you like to proceed with a specific value for \(A\), or would you like more details on how to solve this equation?

Here are some related questions you might find helpful:

1. How do you solve trigonometric equations when identities are involved?
2. What are the common trigonometric identities used in solving equations?
3. How can we simplify equations involving sine and cosine?
4. What is the importance of factoring in solving equations?
5. How do we handle multiple solutions in trigonometric equations?

**Tip:** When solving trigonometric equations, always check for extraneous solutions by substituting back into the original equation!

This guide explains how to solve the trigonometric equation A = sin(x)/sin(pi/3 - x) using the sine subtraction identity and algebraic methods. Explore step-by-step solutions involving trigonometric identities and equation-solving techniques. Suitable for students in Grades 10-12.

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