Given the equation:

\[
\sin 30^\circ \cdot \cos \theta + \cos 30^\circ = 2 \sin 60^\circ
\]

Let's start by substituting the values of the trigonometric functions:

- \(\sin 30^\circ = \frac{1}{2}\)
- \(\cos 30^\circ = \frac{\sqrt{3}}{2}\)
- \(\sin 60^\circ = \frac{\sqrt{3}}{2}\)

Substituting these values into the equation:

\[
\frac{1}{2} \cdot \cos \theta + \frac{\sqrt{3}}{2} = 2 \cdot \frac{\sqrt{3}}{2}
\]

Simplify the right-hand side:

\[
\frac{1}{2} \cdot \cos \theta + \frac{\sqrt{3}}{2} = \sqrt{3}
\]

Subtract \(\frac{\sqrt{3}}{2}\) from both sides:

\[
\frac{1}{2} \cos \theta = \sqrt{3} - \frac{\sqrt{3}}{2}
\]

Combine the terms on the right:

\[
\sqrt{3} - \frac{\sqrt{3}}{2} = \frac{2\sqrt{3}}{2} - \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}
\]

So, the equation simplifies to:

\[
\frac{1}{2} \cos \theta = \frac{\sqrt{3}}{2}
\]

Multiply both sides by 2:

\[
\cos \theta = \sqrt{3}
\]

Since \(\cos \theta\) cannot equal \(\sqrt{3}\) (as \(\cos \theta\) values range between -1 and 1), there is no real value of \(\theta\) that satisfies this equation. 

### Key Takeaways:
- The values of trigonometric functions should be within their defined range.
- Ensure to simplify expressions accurately to detect inconsistencies.

Would you like more details, have any questions, or need assistance with a similar problem?

### Related Questions:
1. What are the ranges of sine and cosine functions?
2. How do we solve trigonometric equations involving multiple angles?
3. What are some common mistakes when solving trigonometric equations?
4. How do transformations affect trigonometric graphs?
5. What are the implications of having no real solution in a trigonometric equation?

**Tip:** Always check the domain and range constraints when solving trigonometric equations to ensure valid solutions.

Learn how to solve the trigonometric equation sin 30 cos θ + cos 30 = 2 sin 60 step-by-step. This problem involves trigonometric identities and equations, with detailed explanations and key takeaways on trigonometric functions and their ranges. Suitable for high school students studying trigonometry.

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