Find the value of `(sqrt(3) cos 23^(@) - sin 23^(@))/(2)`

Options

A. `cos 53^(@)`
B. `sin 53^(@)`
C. `tan 53^(@)`
D. 1

Answer

To find the value of \(\frac{\sqrt{3} \cos 23^\circ - \sin 23^\circ}{2}\), we can follow these steps:

### Step 1: Rewrite the expression
We can separate the terms in the numerator:
\[ \frac{\sqrt{3} \cos 23^\circ - \sin 23^\circ}{2} = \frac{\sqrt{3}}{2} \cos 23^\circ - \frac{1}{2} \sin 23^\circ \]

### Step 2: Identify trigonometric values
Recall that:
\[ \frac{\sqrt{3}}{2} = \sin 60^\circ \quad \text{and} \quad \frac{1}{2} = \cos 60^\circ \]
Thus, we can rewrite the expression as:
\[ \sin 60^\circ \cos 23^\circ - \cos 60^\circ \sin 23^\circ \]

### Step 3: Apply the sine subtraction formula
The expression \(\sin A \cos B - \cos A \sin B\) can be rewritten using the sine subtraction formula:
\[ \sin(A - B) \]
In our case, we have:
\[ \sin(60^\circ - 23^\circ) = \sin 37^\circ \]

### Step 4: Final expression
Thus, we can simplify the original expression to:
\[ \frac{\sqrt{3} \cos 23^\circ - \sin 23^\circ}{2} = \sin 37^\circ \]

### Step 5: Alternative representation
We can also express \(\sin 37^\circ\) in terms of cosine:
\[ \sin 37^\circ = \cos(90^\circ - 37^\circ) = \cos 53^\circ \]

### Conclusion
Therefore, the final answer is:
\[ \frac{\sqrt{3} \cos 23^\circ - \sin 23^\circ}{2} = \cos 53^\circ \]

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