If `tan A = 1/2, ` then `tan 3A=`

Options

A. ` 9/2`
B. `11/2`
C. `7/2`
D. `-1/2`

Answer

To find \( \tan 3A \) given that \( \tan A = \frac{1}{2} \), we will use the trigonometric identity for \( \tan 3A \):

\[ \tan 3A = \frac{3 \tan A - \tan^3 A}{1 - 3 \tan^2 A} \]

### Step-by-Step Solution:

**Step 1: Identify \( \tan A \)**
Given \( \tan A = \frac{1}{2} \).

**Step 2: Calculate \( \tan^2 A \)**
\[ \tan^2 A = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

**Step 3: Calculate \( \tan^3 A \)**
\[ \tan^3 A = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]

**Step 4: Substitute values into the formula for \( \tan 3A \)**
Now, we substitute \( \tan A \), \( \tan^2 A \), and \( \tan^3 A \) into the formula:
\[ \tan 3A = \frac{3 \cdot \frac{1}{2} - \frac{1}{8}}{1 - 3 \cdot \frac{1}{4}} \]

**Step 5: Simplify the numerator**
Calculating the numerator:
\[ 3 \cdot \frac{1}{2} = \frac{3}{2} \]
So,
\[ \frac{3}{2} - \frac{1}{8} = \frac{12}{8} - \frac{1}{8} = \frac{11}{8} \]

**Step 6: Simplify the denominator**
Calculating the denominator:
\[ 1 - 3 \cdot \frac{1}{4} = 1 - \frac{3}{4} = \frac{1}{4} \]

**Step 7: Combine the results**
Now, substituting back into the formula:
\[ \tan 3A = \frac{\frac{11}{8}}{\frac{1}{4}} = \frac{11}{8} \cdot 4 = \frac{11}{2} \]

### Final Answer:
\[ \tan 3A = \frac{11}{2} \]

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