What is the value of `(sin30^(@)+cos30^(@))/(cos30^(@)-sin30^(@))`?

Options

A. `2-sqrt(3)`
B. `sqrt(3)`
C. 1
D. `-(2-sqrt(3))`

Answer

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

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

### Step 2: Substitute the values into the expression
Now, substituting these values into the expression:
\[ \frac{\sin 30^\circ + \cos 30^\circ}{\cos 30^\circ - \sin 30^\circ} = \frac{\frac{1}{2} + \frac{\sqrt{3}}{2}}{\frac{\sqrt{3}}{2} - \frac{1}{2}} \]

### Step 3: Simplify the numerator and denominator
The numerator becomes:
\[ \frac{1}{2} + \frac{\sqrt{3}}{2} = \frac{1 + \sqrt{3}}{2} \]
The denominator becomes:
\[ \frac{\sqrt{3}}{2} - \frac{1}{2} = \frac{\sqrt{3} - 1}{2} \]

### Step 4: Write the expression with simplified numerator and denominator
Now, we can rewrite the expression as:
\[ \frac{\frac{1 + \sqrt{3}}{2}}{\frac{\sqrt{3} - 1}{2}} \]

### Step 5: Cancel the common factor of \(\frac{1}{2}\)
When we simplify this, the \(\frac{1}{2}\) in the numerator and denominator cancels out:
\[ \frac{1 + \sqrt{3}}{\sqrt{3} - 1} \]

### Step 6: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, \(\sqrt{3} + 1\):
\[ \frac{(1 + \sqrt{3})(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} \]

### Step 7: Simplify the denominator
The denominator simplifies as follows:
\[ (\sqrt{3})^2 - (1)^2 = 3 - 1 = 2 \]

### Step 8: Expand the numerator
Now, expanding the numerator:
\[ (1 + \sqrt{3})(\sqrt{3} + 1) = 1 \cdot \sqrt{3} + 1 \cdot 1 + \sqrt{3} \cdot \sqrt{3} + \sqrt{3} \cdot 1 = \sqrt{3} + 1 + 3 + \sqrt{3} = 4 + 2\sqrt{3} \]

### Step 9: Combine the results
So we have:
\[ \frac{4 + 2\sqrt{3}}{2} \]
Dividing both terms in the numerator by 2 gives:
\[ 2 + \sqrt{3} \]

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