`1/3 (sqrt3 cos 23^(@) - sin 23^(@))=`

Options

A. `cos 43^(@) `
B. ` cos 7^(@)`
C. `1/4 cos 53^(@)`
D. ` 2/3 cos 53^(@)`

Answer

To solve the equation \( \frac{1}{3} (\sqrt{3} \cos 23^\circ - \sin 23^\circ) \), we will follow these steps:

### Step 1: Factor out the common term
We start with the expression:
\[ \frac{1}{3} (\sqrt{3} \cos 23^\circ - \sin 23^\circ) \]

### Step 2: Identify the cosine of a compound angle
We can rewrite the expression inside the parentheses using the cosine of a sum formula. The cosine of a sum is given by:
\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]
We want to express \( \sqrt{3} \cos 23^\circ - \sin 23^\circ \) in a similar form.

### Step 3: Recognize the coefficients
Notice that \( \sqrt{3} \) is the coefficient of \( \cos 23^\circ \) and \( -1 \) is the coefficient of \( \sin 23^\circ \). This suggests that we can relate it to the cosine of a specific angle.

### Step 4: Rewrite using known angles
We know that:
\[ \sqrt{3} = 2 \cos 30^\circ \quad \text{and} \quad -1 = -2 \sin 30^\circ \]
Thus, we can rewrite the expression as:
\[ \sqrt{3} \cos 23^\circ - \sin 23^\circ = 2 \left( \cos 30^\circ \cos 23^\circ - \sin 30^\circ \sin 23^\circ \right) \]

### Step 5: Apply the cosine of a sum formula
Using the cosine of a sum formula:
\[ \sqrt{3} \cos 23^\circ - \sin 23^\circ = 2 \cos(30^\circ + 23^\circ) = 2 \cos(53^\circ) \]

### Step 6: Substitute back into the original expression
Now substituting back into our original equation:
\[ \frac{1}{3} (2 \cos 53^\circ) = \frac{2}{3} \cos 53^\circ \]

### Final Result
Thus, the simplified expression is:
\[ \frac{2}{3} \cos 53^\circ \]