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

Options

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

Answer

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

### Step 1: Find the value of \( \sin 45^\circ \)
The value of \( \sin 45^\circ \) is known to be:
\[ \sin 45^\circ = \frac{1}{\sqrt{2}} \]

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

### Step 3: Find a common denominator
To simplify the expression, we need a common denominator. The common denominator between \( \sqrt{2} \) and \( 1 \) is \( \sqrt{2} \):
\[ \frac{1}{\sqrt{2}} - \sqrt{3} = \frac{1}{\sqrt{2}} - \frac{\sqrt{3} \cdot \sqrt{2}}{\sqrt{2}} = \frac{1 - \sqrt{6}}{\sqrt{2}} \]

### Step 4: Final expression
Thus, the expression simplifies to:
\[ \frac{1 - \sqrt{6}}{\sqrt{2}} \]

### Step 5: Write the final answer
The final answer can be expressed as:
\[ 1 - \frac{\sqrt{6}}{\sqrt{2}} = \frac{1 - \sqrt{6}}{\sqrt{2}} \]

### Conclusion
The value of \( \sin 45^\circ - \sqrt{3} \) is:
\[ \frac{1 - \sqrt{6}}{\sqrt{2}} \]

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