In `triangle`ABC, a = 2, b = 3,c = 4 , then the value of cos A is

Options

A. `21/24`
B. `21/12`
C. `24/21`
D. `24/12`

Answer

To find the value of cos A in triangle ABC with sides a = 2, b = 3, and c = 4, we will use the cosine rule. The cosine rule states that:

\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]

### Step-by-step solution:

1. **Identify the sides**: We have the sides of the triangle as:
- \( a = 2 \)
- \( b = 3 \)
- \( c = 4 \)

2. **Substitute the values into the cosine rule formula**:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
Substituting the values:
\[ \cos A = \frac{3^2 + 4^2 - 2^2}{2 \cdot 3 \cdot 4} \]

3. **Calculate \( b^2 \), \( c^2 \), and \( a^2 \)**:
- \( b^2 = 3^2 = 9 \)
- \( c^2 = 4^2 = 16 \)
- \( a^2 = 2^2 = 4 \)

4. **Substitute the squared values back into the equation**:
\[ \cos A = \frac{9 + 16 - 4}{2 \cdot 3 \cdot 4} \]

5. **Perform the addition and subtraction in the numerator**:
\[ 9 + 16 = 25 \]
\[ 25 - 4 = 21 \]

6. **Calculate the denominator**:
\[ 2 \cdot 3 \cdot 4 = 24 \]

7. **Now substitute back into the equation**:
\[ \cos A = \frac{21}{24} \]

8. **Simplify the fraction**:
\[ \cos A = \frac{21 \div 3}{24 \div 3} = \frac{7}{8} \]

### Final Answer:
\[ \cos A = \frac{7}{8} \]