What is the value of `(1//sqrt3 + sin 45^@)` ?

Options

A. `sqrt3+2`
B. `(sqrt2+sqrt3)//sqrt6`
C. `(1+sqrt6)//sqrt3`
D. `(4+sqrt3)//25`

Answer

To solve the expression \( \frac{1}{\sqrt{3}} + \sin 45^\circ \), we can follow these steps:

### Step 1: Identify the value of \( \sin 45^\circ \)
We know that:
\[ \sin 45^\circ = \frac{1}{\sqrt{2}} \]

### Step 2: Rewrite the expression
Now we can rewrite the original expression by substituting the value of \( \sin 45^\circ \):
\[ \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{2}} \]

### Step 3: Find a common denominator
To add these fractions, we need a common denominator. The least common multiple (LCM) of \( \sqrt{3} \) and \( \sqrt{2} \) is \( \sqrt{6} \). Therefore, we can rewrite each fraction:
\[ \frac{1}{\sqrt{3}} = \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{3}} = \frac{\sqrt{2}}{\sqrt{6}} \]
\[ \frac{1}{\sqrt{2}} = \frac{\sqrt{3}}{\sqrt{3} \cdot \sqrt{2}} = \frac{\sqrt{3}}{\sqrt{6}} \]

### Step 4: Combine the fractions
Now we can add the two fractions:
\[ \frac{\sqrt{2}}{\sqrt{6}} + \frac{\sqrt{3}}{\sqrt{6}} = \frac{\sqrt{2} + \sqrt{3}}{\sqrt{6}} \]

### Step 5: Final expression
Thus, the value of \( \frac{1}{\sqrt{3}} + \sin 45^\circ \) simplifies to:
\[ \frac{\sqrt{2} + \sqrt{3}}{\sqrt{6}} \]