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.

Learn how to evaluate the trigonometric expression 2sin(pi/6) + sqrt(3)cos(pi/6) step-by-step with detailed solutions. This problem involves applying trigonometric identities and simplifying trigonometric functions. Suitable for students in Grades 10-12.

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