What is the value of `sqrt(2) sec 45^(@)+(1)/sqrt(3) tan 30^(@)`

Options

A. `(1+sqrt(3))/(2)`
B. `(1+3sqrt(2))/sqrt(3)`
C. `7/3`
D. `(3+2sqrt(2))/(3sqrt(2))`

Answer

To solve the expression \( \sqrt{2} \sec 45^\circ + \frac{1}{\sqrt{3}} \tan 30^\circ \), we will follow these steps:

### Step 1: Find the value of \( \sec 45^\circ \)
The secant function is the reciprocal of the cosine function. Therefore, we have:
\[ \sec 45^\circ = \frac{1}{\cos 45^\circ} \]
Since \( \cos 45^\circ = \frac{1}{\sqrt{2}} \), we can calculate:
\[ \sec 45^\circ = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]

### Step 2: Find the value of \( \tan 30^\circ \)
The tangent function is the ratio of the sine function to the cosine function. Thus, we have:
\[ \tan 30^\circ = \frac{\sin 30^\circ}{\cos 30^\circ} \]
We know that \( \sin 30^\circ = \frac{1}{2} \) and \( \cos 30^\circ = \frac{\sqrt{3}}{2} \). Therefore:
\[ \tan 30^\circ = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \]

### Step 3: Substitute the values into the expression
Now we can substitute the values we found into the original expression:
\[ \sqrt{2} \sec 45^\circ + \frac{1}{\sqrt{3}} \tan 30^\circ = \sqrt{2} \cdot \sqrt{2} + \frac{1}{\sqrt{3}} \cdot \frac{1}{\sqrt{3}} \]

### Step 4: Simplify the expression
Calculating each term:
\[ \sqrt{2} \cdot \sqrt{2} = 2 \]
\[ \frac{1}{\sqrt{3}} \cdot \frac{1}{\sqrt{3}} = \frac{1}{3} \]
Now combine these results:
\[ 2 + \frac{1}{3} \]

### Step 5: Find a common denominator and add
To add \( 2 \) and \( \frac{1}{3} \), we convert \( 2 \) into a fraction with a denominator of \( 3 \):
\[ 2 = \frac{6}{3} \]
Now add:
\[ \frac{6}{3} + \frac{1}{3} = \frac{7}{3} \]

### Final Answer
Thus, the value of \( \sqrt{2} \sec 45^\circ + \frac{1}{\sqrt{3}} \tan 30^\circ \) is:
\[ \frac{7}{3} \]