If sin B = 0.5, find the value of 3 cos `B-4cos^(3)B`

Answer

To solve the problem, we start with the given information that \( \sin B = 0.5 \). We need to find the value of \( 3 \cos B - 4 \cos^3 B \).

### Step 1: Find \( \cos B \)
Since we know \( \sin B = 0.5 \), we can use the Pythagorean identity:

\[ \sin^2 B + \cos^2 B = 1 \]

Substituting \( \sin B \):

\[ (0.5)^2 + \cos^2 B = 1 \]

Calculating \( (0.5)^2 \):

\[ 0.25 + \cos^2 B = 1 \]

Now, isolate \( \cos^2 B \):

\[ \cos^2 B = 1 - 0.25 = 0.75 \]

Taking the square root to find \( \cos B \):

\[ \cos B = \sqrt{0.75} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

### Step 2: Substitute \( \cos B \) into the expression
Now that we have \( \cos B = \frac{\sqrt{3}}{2} \), we can substitute this value into the expression \( 3 \cos B - 4 \cos^3 B \).

First, we 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 \( \cos B \) and \( \cos^3 B \) into the expression:

\[ 3 \cos B - 4 \cos^3 B = 3 \left(\frac{\sqrt{3}}{2}\right) - 4 \left(\frac{3\sqrt{3}}{8}\right) \]

Calculating \( 3 \cos B \):

\[ 3 \cos B = 3 \cdot \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2} \]

Calculating \( 4 \cos^3 B \):

\[ 4 \cos^3 B = 4 \cdot \frac{3\sqrt{3}}{8} = \frac{12\sqrt{3}}{8} = \frac{3\sqrt{3}}{2} \]

### Step 3: Combine the results
Now, we combine the results:

\[ 3 \cos B - 4 \cos^3 B = \frac{3\sqrt{3}}{2} - \frac{3\sqrt{3}}{2} = 0 \]

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

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