Let's evaluate the given expression:

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

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

$\frac{2 \times \frac{1}{2}}{6} + \frac{\sqrt{3} \times \frac{\sqrt{3}}{2}}{6}$

Step 3: Simplify the expression

The first term:

$\frac{2 \times \frac{1}{2}}{6} = \frac{1}{6}$

The second term:

$\frac{\sqrt{3} \times \frac{\sqrt{3}}{2}}{6} = \frac{\frac{3}{2}}{6} = \frac{3}{12} = \frac{1}{4}$

Step 4: Combine the terms

$\frac{1}{6} + \frac{1}{4}$

To add these fractions, find a common denominator:

$\frac{1}{6} = \frac{2}{12}, \quad \frac{1}{4} = \frac{3}{12}$

$\frac{2}{12} + \frac{3}{12} = \frac{5}{12}$

Final Answer

The expression simplifies to $\frac{5}{12}$.

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Would you like more details or have any questions?

Here are some related questions to consider:

1. How do you calculate $\sin$ and $\cos$ for different angles?
2. What is the importance of finding a common denominator?
3. How would the expression change if $\pi/6$ was replaced with a different angle?
4. Can you simplify the expression using another method?
5. How do these trigonometric identities relate to the unit circle?

Tip: When dealing with trigonometric functions, it’s often helpful to remember the key angles and their sine and cosine values for quick reference.