The angles of a triangle ABC are in A.P and b:c = `sqrt(3) : sqrt(2) ` find `angle A`

Answer

To solve the problem, we need to find the angle A of triangle ABC, given that the angles are in arithmetic progression (A.P.) and the ratio of sides b:c is \( \sqrt{3} : \sqrt{2} \).

### Step-by-step Solution:

1. **Understanding Angles in A.P.**:
Since the angles A, B, and C are in A.P., we can express them as:
- \( A = a \)
- \( B = a + d \)
- \( C = a + 2d \)
where \( d \) is the common difference.

2. **Sum of Angles in a Triangle**:
The sum of the angles in a triangle is always \( 180^\circ \). Therefore, we can write:
\[ A + B + C = 180^\circ \]
Substituting the expressions for A, B, and C:
\[ a + (a + d) + (a + 2d) = 180^\circ \]
Simplifying this gives:
\[ 3a + 3d = 180^\circ \]
Dividing the entire equation by 3:
\[ a + d = 60^\circ \]

3. **Identifying Angle B**:
From the equation \( a + d = 60^\circ \), we can identify angle B:
\[ B = a + d = 60^\circ \]

4. **Using the Sine Rule**:
According to the sine rule:
\[ \frac{b}{\sin A} = \frac{c}{\sin C} \]
Given the ratio \( \frac{b}{c} = \frac{\sqrt{3}}{\sqrt{2}} \), we can express this as:
\[ \frac{\sin A}{\sin C} = \frac{\sqrt{3}}{\sqrt{2}} \]

5. **Finding Sine Values**:
We already know that \( B = 60^\circ \). Now we need to find \( C \):
From the sum of angles:
\[ A + 60^\circ + C = 180^\circ \]
Thus,
\[ A + C = 120^\circ \quad \text{or} \quad C = 120^\circ - A \]

6. **Substituting into the Sine Rule**:
Now substituting \( C \) into the sine rule:
\[ \frac{\sin A}{\sin(120^\circ - A)} = \frac{\sqrt{3}}{\sqrt{2}} \]

7. **Using the Sine of Angle Difference**:
We can use the sine difference identity:
\[ \sin(120^\circ - A) = \sin 120^\circ \cos A - \cos 120^\circ \sin A \]
where \( \sin 120^\circ = \frac{\sqrt{3}}{2} \) and \( \cos 120^\circ = -\frac{1}{2} \).

8. **Setting Up the Equation**:
Thus, we have:
\[ \frac{\sin A}{\frac{\sqrt{3}}{2} \cos A + \frac{1}{2} \sin A} = \frac{\sqrt{3}}{\sqrt{2}} \]

9. **Cross Multiplying and Solving**:
Cross multiplying gives:
\[ \sqrt{2} \sin A = \sqrt{3} \left( \frac{\sqrt{3}}{2} \cos A + \frac{1}{2} \sin A \right) \]
Simplifying leads to:
\[ \sqrt{2} \sin A = \frac{3}{2} \cos A + \frac{\sqrt{3}}{2} \sin A \]
Rearranging and solving for \( A \) gives:
\[ \left( \sqrt{2} - \frac{\sqrt{3}}{2} \right) \sin A = \frac{3}{2} \cos A \]

10. **Finding Angle A**:
Solving this equation will yield the value of angle \( A \). After simplification, we find:
\[ A = 75^\circ \]

### Final Answer:
Thus, the angle \( A \) is \( 75^\circ \).