$$ \sin\left(\theta-\frac{\pi}{3}\right) $$$$ =\frac12 $$, $$ \theta\in\text{  \lbrack}0,\text{ }2\pi] $$

To solve the equation 

\[
\sin\left(\theta - \frac{\pi}{3}\right) = \frac{1}{2},
\]

where $\theta \in [0, 2\pi]$, we will find the general solutions for $\theta - \frac{\pi}{3}$ and then solve for $\theta$.

### Step 1: Solve the General Equation for Sine

The sine function equals $\frac{1}{2}$ at two points in the interval $[0, 2\pi]$:

\[
\sin x = \frac{1}{2} \implies x = \frac{\pi}{6} \quad \text{or} \quad x = \pi - \frac{\pi}{6} = \frac{5\pi}{6}.
\]

Therefore, for the equation $\sin\left(\theta - \frac{\pi}{3}\right) = \frac{1}{2}$, we have:

\[
\theta - \frac{\pi}{3} = \frac{\pi}{6} \quad \text{or} \quad \theta - \frac{\pi}{3} = \frac{5\pi}{6}.
\]

### Step 2: Solve for $\theta$

1. **First Case:**

   \[
   \theta - \frac{\pi}{3} = \frac{\pi}{6}
   \]

   Adding $\frac{\pi}{3}$ to both sides:

   \[
   \theta = \frac{\pi}{6} + \frac{\pi}{3} = \frac{\pi}{6} + \frac{2\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2}.
   \]

2. **Second Case:**

   \[
   \theta - \frac{\pi}{3} = \frac{5\pi}{6}
   \]

   Adding $\frac{\pi}{3}$ to both sides:

   \[
   \theta = \frac{5\pi}{6} + \frac{\pi}{3} = \frac{5\pi}{6} + \frac{2\pi}{6} = \frac{7\pi}{6}.
   \]

### Step 3: Verify that Solutions are within the Interval $[0, 2\pi]$

Both solutions $\frac{\pi}{2}$ and $\frac{7\pi}{6}$ lie within the interval $[0, 2\pi]$.

### Conclusion

The solutions to the equation are:

\[
\theta = \frac{\pi}{2} \quad \text{and} \quad \theta = \frac{7\pi}{6}.
\]

Would you like more details or have any questions?

### Related Questions:

1. How do you solve trigonometric equations involving cosine or tangent?
2. What are the general solutions for $\sin x = \frac{1}{2}$?
3. How can you solve trigonometric equations with multiple angles, like $\sin(2\theta) = \frac{1}{2}$?
4. What is the process for solving equations involving inverse trigonometric functions?
5. How do you solve trigonometric equations involving phase shifts, like $\cos(\theta + \alpha) = \frac{1}{2}$?

**Tip:** When solving trigonometric equations, it's helpful to visualize the unit circle and remember the key angles where sine, cosine, and tangent take on specific values.

Learn how to solve the trigonometric equation $ \sin(\theta - \frac{\pi}{3}) = \frac{1}{2} $ with detailed steps. Discover the solutions within the interval $ [0, 2\pi] $ and verify them. Suitable for high school students studying trigonometry.

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