What is the value of `((1)/(sqrt(3))-sin45^(@))`?

Options

A. `(1)/(2sqrt(3))`
B. `(sqrt(2)-sqrt(3))/(sqrt(6))`
C. `1-sqrt(2)`
D. `((4-sqrt(3)))/(2)`

Answer

To solve the question \( \frac{1}{\sqrt{3}} - \sin 45^\circ \), we will follow these steps:

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

### Step 2: Substitute the value of \( \sin 45^\circ \) into the expression
Now we substitute \( \sin 45^\circ \) into the expression:
\[ \frac{1}{\sqrt{3}} - \sin 45^\circ = \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{2}} \]

### Step 3: Find a common denominator
To subtract these fractions, we need a common denominator. The least common multiple of \( \sqrt{3} \) and \( \sqrt{2} \) is \( \sqrt{6} \). We can rewrite each fraction with this common denominator:
\[ \frac{1}{\sqrt{3}} = \frac{\sqrt{2}}{\sqrt{6}} \quad \text{and} \quad \frac{1}{\sqrt{2}} = \frac{\sqrt{3}}{\sqrt{6}} \]

### Step 4: Rewrite the expression with the common denominator
Now we can rewrite the expression:
\[ \frac{\sqrt{2}}{\sqrt{6}} - \frac{\sqrt{3}}{\sqrt{6}} = \frac{\sqrt{2} - \sqrt{3}}{\sqrt{6}} \]

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