Let's simplify the given expression:

$2\sin\left(\frac{\pi}{6}\right) + \sqrt{3}\cos\left(\frac{\pi}{6}\right)$

Step 1: Evaluate $\sin\left(\frac{\pi}{6}\right)$ and $\cos\left(\frac{\pi}{6}\right)$

$\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$ $\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$

Step 2: Substitute these values into the expression

$2 \times \frac{1}{2} + \sqrt{3} \times \frac{\sqrt{3}}{2}$

Step 3: Simplify the expression

The first term:

$2 \times \frac{1}{2} = 1$

The second term:

$\sqrt{3} \times \frac{\sqrt{3}}{2} = \frac{3}{2}$

Step 4: Combine the terms

$1 + \frac{3}{2}$

To combine these, convert 1 to $\frac{2}{2}$:

$\frac{2}{2} + \frac{3}{2} = \frac{5}{2}$

Final Answer

The expression simplifies to $\frac{5}{2}$.

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Do you want more details or have any questions?

Here are some related questions to explore:

1. What are some other key angles and their sine and cosine values?
2. How does this expression relate to the unit circle?
3. What happens if you change the angle from $\frac{\pi}{6}$ to another value?
4. How can trigonometric identities simplify expressions like this one?
5. What is the significance of $\frac{\pi}{6}$ in trigonometry?

Tip: Knowing key trigonometric values can help you quickly simplify expressions like this.