What is the value of `(tan30^(@)+(sqrt(3))/(2))`?

Options

A. `((4+sqrt(3)))/(2)`
B. `(5)/(2sqrt(3))`
C. `((4+sqrt(3)))/(2sqrt(3))`
D. `((sqrt(3)+1))/(sqrt(3))`

Answer

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

### Step-by-Step Solution:

1. **Calculate \( \tan(30^\circ) \)**:
The value of \( \tan(30^\circ) \) is known to be \( \frac{1}{\sqrt{3}} \).

\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]

2. **Write the expression**:
Now substitute \( \tan(30^\circ) \) into the expression:

\[ \tan(30^\circ) + \frac{\sqrt{3}}{2} = \frac{1}{\sqrt{3}} + \frac{\sqrt{3}}{2} \]

3. **Find a common denominator**:
The denominators are \( \sqrt{3} \) and \( 2 \). The least common multiple (LCM) of these denominators is \( 2\sqrt{3} \).

4. **Convert each term to have the common denominator**:
- For \( \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{2}{2\sqrt{3}} \]

- For \( \frac{\sqrt{3}}{2} \):
\[ \frac{\sqrt{3}}{2} = \frac{\sqrt{3} \cdot \sqrt{3}}{2 \cdot \sqrt{3}} = \frac{3}{2\sqrt{3}} \]

5. **Combine the fractions**:
Now we can add the two fractions:

\[ \frac{2}{2\sqrt{3}} + \frac{3}{2\sqrt{3}} = \frac{2 + 3}{2\sqrt{3}} = \frac{5}{2\sqrt{3}} \]

6. **Final answer**:
Thus, the value of \( \tan(30^\circ) + \frac{\sqrt{3}}{2} \) is:

\[ \frac{5}{2\sqrt{3}} \]