What is the value of `((1)/(2)-sec30^(@))`?

Options

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

Answer

To find the value of \(\frac{1}{2} - \sec 30^\circ\), we will follow these steps:

### Step 1: Calculate \(\sec 30^\circ\)
The secant function is the reciprocal of the cosine function. Therefore, we first need to find \(\cos 30^\circ\).

\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]

Now, we can find \(\sec 30^\circ\):

\[ \sec 30^\circ = \frac{1}{\cos 30^\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]

### Step 2: Substitute \(\sec 30^\circ\) into the expression
Now we substitute \(\sec 30^\circ\) back into the original expression:

\[ \frac{1}{2} - \sec 30^\circ = \frac{1}{2} - \frac{2}{\sqrt{3}} \]

### Step 3: Find a common denominator
To subtract these fractions, we need a common denominator. The common denominator between \(2\) and \(\sqrt{3}\) is \(2\sqrt{3}\).

Rewriting the fractions with the common denominator:

\[ \frac{1}{2} = \frac{1 \cdot \sqrt{3}}{2 \cdot \sqrt{3}} = \frac{\sqrt{3}}{2\sqrt{3}} \]

Now, rewrite \(\frac{2}{\sqrt{3}}\):

\[ \frac{2}{\sqrt{3}} = \frac{2 \cdot 2}{\sqrt{3} \cdot 2} = \frac{4}{2\sqrt{3}} \]

### Step 4: Subtract the fractions
Now we can subtract the two fractions:

\[ \frac{\sqrt{3}}{2\sqrt{3}} - \frac{4}{2\sqrt{3}} = \frac{\sqrt{3} - 4}{2\sqrt{3}} \]

### Final Answer
Thus, the value of \(\frac{1}{2} - \sec 30^\circ\) is:

\[ \frac{\sqrt{3} - 4}{2\sqrt{3}} \]

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