What is the value of `(sin 30^@ + cos 30^@)/(cos 30^@ - sin 30^@)`

Options

A. `2 - sqrt3`
B. `2+ sqrt3`
C. 1
D. `-(-2-sqrt3)`

Answer

To solve the expression \((\sin 30^\circ + \cos 30^\circ) / (\cos 30^\circ - \sin 30^\circ)\), we will follow these steps:

### Step 1: Find the values of \(\sin 30^\circ\) and \(\cos 30^\circ\).
We know:
\[ \sin 30^\circ = \frac{1}{2} \]
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]

### Step 2: Substitute these values into the expression.
Substituting the values we found:
\[ \frac{\sin 30^\circ + \cos 30^\circ}{\cos 30^\circ - \sin 30^\circ} = \frac{\frac{1}{2} + \frac{\sqrt{3}}{2}}{\frac{\sqrt{3}}{2} - \frac{1}{2}} \]

### Step 3: Simplify the numerator and denominator.
**Numerator:**
\[ \frac{1}{2} + \frac{\sqrt{3}}{2} = \frac{1 + \sqrt{3}}{2} \]

**Denominator:**
\[ \frac{\sqrt{3}}{2} - \frac{1}{2} = \frac{\sqrt{3} - 1}{2} \]

### Step 4: Rewrite the expression.
Now we can rewrite the expression:
\[ \frac{\frac{1 + \sqrt{3}}{2}}{\frac{\sqrt{3} - 1}{2}} = \frac{1 + \sqrt{3}}{\sqrt{3} - 1} \]

### Step 5: Rationalize the denominator.
To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(1 + \sqrt{3})(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} \]

### Step 6: Simplify the numerator and denominator.
**Numerator:**
\[ (1 + \sqrt{3})(\sqrt{3} + 1) = 1 \cdot \sqrt{3} + 1 \cdot 1 + \sqrt{3} \cdot \sqrt{3} + \sqrt{3} \cdot 1 = \sqrt{3} + 1 + 3 + \sqrt{3} = 4 + 2\sqrt{3} \]

**Denominator:**
\[ (\sqrt{3} - 1)(\sqrt{3} + 1) = \sqrt{3}^2 - 1^2 = 3 - 1 = 2 \]

### Step 7: Combine the results.
Now we have:
\[ \frac{4 + 2\sqrt{3}}{2} = 2 + \sqrt{3} \]

### Final Answer:
Thus, the value of the expression \((\sin 30^\circ + \cos 30^\circ) / (\cos 30^\circ - \sin 30^\circ)\) is:
\[ \boxed{2 + \sqrt{3}} \]