The sides a, b, c of a triangle ABC are in arithmetic progression and 'a' is the smallest side. What is cos A equal to ?

Options

A. `(3c - 4b)/(2c)`
B. `(3c - 4b)/(2b)`
C. `(4c - 3b)/(2c)`
D. `(3b - 4c)/(2c)`

Explanation

To solve the problem, we need to find the value of cos A in triangle ABC, given that the sides a, b, and c are in arithmetic progression (AP) and that 'a' is the smallest side.

### Step-by-Step Solution:

1. **Understanding the Arithmetic Progression**:
Since the sides a, b, and c are in arithmetic progression and a is the smallest side, we can express the sides as:
\[ a = a, \quad b = a + d, \quad c = a + 2d \]
where \(d\) is the common difference.

2. **Using the Cosine Rule**:
The cosine of angle A can be expressed using the cosine rule:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]

3. **Substituting the Values of b and c**:
Substitute \(b\) and \(c\) in terms of \(a\) and \(d\):
\[ b = a + d, \quad c = a + 2d \]
Now, we calculate \(b^2\) and \(c^2\):
\[ b^2 = (a + d)^2 = a^2 + 2ad + d^2 \]
\[ c^2 = (a + 2d)^2 = a^2 + 4ad + 4d^2 \]

4. **Calculating b^2 + c^2 - a^2**:
Now, we compute \(b^2 + c^2 - a^2\):
\[ b^2 + c^2 = (a^2 + 2ad + d^2) + (a^2 + 4ad + 4d^2) = 2a^2 + 6ad + 5d^2 \]
Thus,
\[ b^2 + c^2 - a^2 = 2a^2 + 6ad + 5d^2 - a^2 = a^2 + 6ad + 5d^2 \]

5. **Calculating 2bc**:
Now, we calculate \(2bc\):
\[ 2bc = 2(a + d)(a + 2d) = 2(a^2 + 3ad + 2d^2) \]

6. **Substituting into the Cosine Formula**:
Now we substitute these into the cosine formula:
\[ \cos A = \frac{a^2 + 6ad + 5d^2}{2(a^2 + 3ad + 2d^2)} \]

7. **Simplifying the Expression**:
This expression can be simplified further, but the exact simplification depends on the values of \(a\) and \(d\). However, we can express it in terms of \(a\) and \(d\).

### Final Result:
Thus, the expression for \(\cos A\) in terms of \(a\) and \(d\) is:
\[ \cos A = \frac{a^2 + 6ad + 5d^2}{2(a^2 + 3ad + 2d^2)} \]