Sides of `DeltaABC` are in A.P. If `a lt "min" {b,c}`, then cos A may be equal to

Options

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

Explanation

To solve the problem step by step, we need to analyze the given conditions and derive the values of cos A based on the properties of triangles and arithmetic progressions.

### Step 1: Understand the Given Conditions
We know that the sides of triangle ABC are in Arithmetic Progression (AP). This means that if we denote the sides as a, b, and c, then:
- Case 1: \( a < \min(b, c) \) implies \( b \) and \( c \) are greater than \( a \).
- Case 2: \( b < \min(a, c) \) implies \( a \) and \( c \) are greater than \( b \).

### Step 2: Case 1 - Sides in AP as \( a, b, c \)
If the sides are in AP, we can express this as:
\[ 2b = a + c \]
From the triangle inequality, since \( a < b \) and \( a < c \), we can express \( a \) in terms of \( b \) and \( c \):
\[ a = 2c - b \]

### Step 3: Use the Cosine Rule
The cosine of angle A can be calculated using the cosine rule:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]
Substituting \( a = 2c - b \):
\[ \cos A = \frac{b^2 + c^2 - (2c - b)^2}{2bc} \]

### Step 4: Simplify the Expression
Expanding \( (2c - b)^2 \):
\[ (2c - b)^2 = 4c^2 - 4bc + b^2 \]
Now substituting back into the cosine formula:
\[ \cos A = \frac{b^2 + c^2 - (4c^2 - 4bc + b^2)}{2bc} \]
This simplifies to:
\[ \cos A = \frac{b^2 + c^2 - 4c^2 + 4bc - b^2}{2bc} \]
\[ \cos A = \frac{4bc - 3c^2}{2bc} \]
Factoring out common terms gives:
\[ \cos A = \frac{4b - 3c}{2b} \]

### Step 5: Case 2 - Sides in AP as \( b, a, c \)
In this case, we express the sides as:
\[ 2a = b + c \]
From the triangle inequality, since \( b < a \) and \( b < c \), we can express \( b \) in terms of \( a \) and \( c \):
\[ b = 2a - c \]

### Step 6: Use the Cosine Rule Again
Using the cosine rule again:
\[ \cos A = \frac{a^2 + c^2 - b^2}{2ac} \]
Substituting \( b = 2a - c \):
\[ \cos A = \frac{a^2 + c^2 - (2a - c)^2}{2ac} \]

### Step 7: Simplify This Expression
Expanding \( (2a - c)^2 \):
\[ (2a - c)^2 = 4a^2 - 4ac + c^2 \]
Substituting back into the cosine formula:
\[ \cos A = \frac{a^2 + c^2 - (4a^2 - 4ac + c^2)}{2ac} \]
This simplifies to:
\[ \cos A = \frac{a^2 + c^2 - 4a^2 + 4ac - c^2}{2ac} \]
\[ \cos A = \frac{4ac - 3a^2}{2ac} \]
Factoring gives:
\[ \cos A = \frac{4c - 3a}{2c} \]

### Final Result
Thus, the possible values for \( \cos A \) are:
1. \( \cos A = \frac{4b - 3c}{2b} \) from Case 1
2. \( \cos A = \frac{4c - 3b}{2c} \) from Case 2