What is the value of `((1)/(2)) sec 30^(@) + sqrt2 tan 60^(@)`?

Options

A. `((1 + 3sqrt2))/(sqrt3)`
B. `((sqrt3 + 2))/(sqrt3)`
C. `sqrt3 + 2`
D. `((sqrt3 + 2))/(2)`

Answer

To solve the expression \(\frac{1}{2} \sec 30^\circ + \sqrt{2} \tan 60^\circ\), we will follow these steps:

### Step 1: Find the values of \(\sec 30^\circ\) and \(\tan 60^\circ\)

1. **Calculate \(\sec 30^\circ\)**:
\[ \sec 30^\circ = \frac{1}{\cos 30^\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]

2. **Calculate \(\tan 60^\circ\)**:
\[ \tan 60^\circ = \sqrt{3} \]

### Step 2: Substitute the values back into the expression

Now substitute the values of \(\sec 30^\circ\) and \(\tan 60^\circ\) into the original expression:
\[ \frac{1}{2} \sec 30^\circ + \sqrt{2} \tan 60^\circ = \frac{1}{2} \cdot \frac{2}{\sqrt{3}} + \sqrt{2} \cdot \sqrt{3} \]

### Step 3: Simplify each term

1. **Simplify the first term**:
\[ \frac{1}{2} \cdot \frac{2}{\sqrt{3}} = \frac{1}{\sqrt{3}} \]

2. **Simplify the second term**:
\[ \sqrt{2} \cdot \sqrt{3} = \sqrt{6} \]

### Step 4: Combine the terms

Now combine the simplified terms:
\[ \frac{1}{\sqrt{3}} + \sqrt{6} \]

### Step 5: Find a common denominator

To combine these fractions, we can express \(\sqrt{6}\) with a common denominator of \(\sqrt{3}\):
\[ \sqrt{6} = \frac{\sqrt{6} \cdot \sqrt{3}}{\sqrt{3}} = \frac{\sqrt{18}}{\sqrt{3}} = \frac{3\sqrt{2}}{\sqrt{3}} \]

Thus, we can rewrite the expression as:
\[ \frac{1 + 3\sqrt{2}}{\sqrt{3}} \]

### Final Result

The final value of the expression \(\frac{1}{2} \sec 30^\circ + \sqrt{2} \tan 60^\circ\) is:
\[ \frac{1 + 3\sqrt{2}}{\sqrt{3}} \]