If `sin B = (1)/(2),` then ` 3 cos B - 4 cos ^(3)B` is equal to
(a) 1
(b) `3/4`
(c) 0
(d) `2 (1)/(2)`

Options

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

Answer

To solve the problem, we need to find the value of \(3 \cos B - 4 \cos^3 B\) given that \(\sin B = \frac{1}{2}\).

### Step-by-Step Solution:

1. **Identify the values of sine and cosine:**
Given \(\sin B = \frac{1}{2}\), we can use the sine definition in a right triangle. In a right triangle, if \(\sin B = \frac{1}{2}\), we can consider the opposite side to be 1 and the hypotenuse to be 2.

2. **Use the Pythagorean theorem to find the adjacent side:**
Using the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2 \]
\[ 2^2 = 1^2 + \text{Adjacent}^2 \]
\[ 4 = 1 + \text{Adjacent}^2 \]
\[ \text{Adjacent}^2 = 4 - 1 = 3 \]
\[ \text{Adjacent} = \sqrt{3} \]

3. **Calculate \(\cos B\):**
Now, we can find \(\cos B\):
\[ \cos B = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{\sqrt{3}}{2} \]

4. **Substitute \(\cos B\) into the expression:**
We need to evaluate \(3 \cos B - 4 \cos^3 B\):
\[ 3 \cos B - 4 \cos^3 B = 3 \left(\frac{\sqrt{3}}{2}\right) - 4 \left(\frac{\sqrt{3}}{2}\right)^3 \]

5. **Calculate \(4 \cos^3 B\):**
First, calculate \(\cos^3 B\):
\[ \cos^3 B = \left(\frac{\sqrt{3}}{2}\right)^3 = \frac{(\sqrt{3})^3}{2^3} = \frac{3\sqrt{3}}{8} \]
Now substitute this back into the expression:
\[ 4 \cos^3 B = 4 \cdot \frac{3\sqrt{3}}{8} = \frac{12\sqrt{3}}{8} = \frac{3\sqrt{3}}{2} \]

6. **Combine the terms:**
Now substitute back into the expression:
\[ 3 \cos B - 4 \cos^3 B = 3 \cdot \frac{\sqrt{3}}{2} - \frac{3\sqrt{3}}{2} \]
\[ = \frac{3\sqrt{3}}{2} - \frac{3\sqrt{3}}{2} = 0 \]

### Final Answer:
Thus, the value of \(3 \cos B - 4 \cos^3 B\) is \(0\).