What is the value of `(cot30^@ + 2//sqrt3)` ?

Options

A. `5//sqrt3`
B. `(1+2sqrt3)//2`
C. `5//6`
D. `2//sqrt3`

Answer

To solve the problem, we need to find the value of \( \cot 30^\circ + \frac{2}{\sqrt{3}} \).

### Step-by-Step Solution:

1. **Find the value of \( \cot 30^\circ \)**:
- The cotangent function is the reciprocal of the tangent function.
- We know that \( \tan 30^\circ = \frac{1}{\sqrt{3}} \).
- Therefore, \( \cot 30^\circ = \frac{1}{\tan 30^\circ} = \sqrt{3} \).

2. **Substitute \( \cot 30^\circ \) into the expression**:
- Now we can substitute the value of \( \cot 30^\circ \) into the expression:
\[ \cot 30^\circ + \frac{2}{\sqrt{3}} = \sqrt{3} + \frac{2}{\sqrt{3}} \]

3. **Combine the terms**:
- To combine \( \sqrt{3} \) and \( \frac{2}{\sqrt{3}} \), we need a common denominator. The common denominator is \( \sqrt{3} \).
- Rewrite \( \sqrt{3} \) as \( \frac{\sqrt{3} \cdot \sqrt{3}}{\sqrt{3}} = \frac{3}{\sqrt{3}} \).
- Now we can add the two fractions:
\[ \frac{3}{\sqrt{3}} + \frac{2}{\sqrt{3}} = \frac{3 + 2}{\sqrt{3}} = \frac{5}{\sqrt{3}} \]

4. **Final Answer**:
- Thus, the value of \( \cot 30^\circ + \frac{2}{\sqrt{3}} \) is \( \frac{5}{\sqrt{3}} \).