If `sin B = (1)/(2),` then `3 cos B - 4 cos^(3)B` is equal to

Options

A. 1
B. `(3)/(4)`
C. 0
D. `2(1)/(2)`

Answer

To solve the problem where \( \sin B = \frac{1}{2} \) and we need to find the value of \( 3 \cos B - 4 \cos^3 B \), we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know that:
\[ \sin^2 B + \cos^2 B = 1 \]
Given \( \sin B = \frac{1}{2} \), we can find \( \sin^2 B \):
\[ \sin^2 B = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
Now, substituting into the Pythagorean identity:
\[ \frac{1}{4} + \cos^2 B = 1 \]

### Step 2: Solve for \( \cos^2 B \)
Rearranging the equation gives:
\[ \cos^2 B = 1 - \frac{1}{4} = \frac{3}{4} \]

### Step 3: Find \( \cos B \)
Taking the square root of both sides, we find:
\[ \cos B = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

### Step 4: Substitute \( \cos B \) into the Expression
Now we substitute \( \cos B \) into the expression \( 3 \cos B - 4 \cos^3 B \):
\[ 3 \cos B - 4 \cos^3 B = 3 \left(\frac{\sqrt{3}}{2}\right) - 4 \left(\frac{\sqrt{3}}{2}\right)^3 \]

### Step 5: Calculate \( \cos^3 B \)
First, calculate \( \cos^3 B \):
\[ \cos^3 B = \left(\frac{\sqrt{3}}{2}\right)^3 = \frac{3\sqrt{3}}{8} \]

### Step 6: Substitute \( \cos^3 B \) back into the Expression
Now substitute this back:
\[ 3 \left(\frac{\sqrt{3}}{2}\right) - 4 \left(\frac{3\sqrt{3}}{8}\right) \]
Calculating the first term:
\[ 3 \left(\frac{\sqrt{3}}{2}\right) = \frac{3\sqrt{3}}{2} \]
Calculating the second term:
\[ 4 \left(\frac{3\sqrt{3}}{8}\right) = \frac{12\sqrt{3}}{8} = \frac{3\sqrt{3}}{2} \]

### Step 7: Combine the Terms
Now we combine the two terms:
\[ \frac{3\sqrt{3}}{2} - \frac{3\sqrt{3}}{2} = 0 \]

### Final Result
Thus, the value of \( 3 \cos B - 4 \cos^3 B \) is:
\[ \boxed{0} \]