If the angles of a triangle ABC are in AP and `b:c=sqrt3:sqrt2` the what is the measure of angle A?

Options

A. A. `30^(@)`
B. B. `45^(@)`
C. C. `60^(@)`
D. D. `75^(@)`

Answer

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

### Step-by-Step Solution:

1. **Understanding the Angles in AP**:
Since the angles of triangle ABC are in AP, we can denote the angles as:
- Angle A = \(a\)
- Angle B = \(a + d\)
- Angle C = \(a + 2d\)
where \(d\) is the common difference.

2. **Using the Sum of Angles in a Triangle**:
The sum of the angles in a triangle is always \(180^\circ\). Therefore, we can write:
\[ a + (a + d) + (a + 2d) = 180^\circ \]
Simplifying this gives:
\[ 3a + 3d = 180^\circ \]
Dividing through by 3:
\[ a + d = 60^\circ \]
This means that angle B (which is \(a + d\)) is \(60^\circ\).

3. **Using the Ratio of Sides**:
We are given that \(b:c = \sqrt{3}:\sqrt{2}\). By the sine rule, we can express this as:
\[ \frac{b}{c} = \frac{\sin A}{\sin C} \]
Thus:
\[ \frac{\sin A}{\sin C} = \frac{\sqrt{3}}{\sqrt{2}} \]

4. **Finding \(\sin A\) and \(\sin C\)**:
We know angle B is \(60^\circ\). Therefore, we can find angle C using the angle sum property:
\[ A + 60^\circ + C = 180^\circ \implies A + C = 120^\circ \implies C = 120^\circ - A \]
Now substituting this into the sine ratio:
\[ \frac{\sin A}{\sin(120^\circ - A)} = \frac{\sqrt{3}}{\sqrt{2}} \]

5. **Using the Sine of a Difference**:
We can use the sine difference identity:
\[ \sin(120^\circ - A) = \sin 120^\circ \cos A - \cos 120^\circ \sin A \]
Knowing that \(\sin 120^\circ = \frac{\sqrt{3}}{2}\) and \(\cos 120^\circ = -\frac{1}{2}\), we have:
\[ \sin(120^\circ - A) = \frac{\sqrt{3}}{2} \cos A + \frac{1}{2} \sin A \]

6. **Setting Up the Equation**:
Now substituting back into the sine ratio:
\[ \sin A = \frac{\sqrt{3}}{\sqrt{2}} \left(\frac{\sqrt{3}}{2} \cos A + \frac{1}{2} \sin A\right) \]
Rearranging gives us an equation in terms of \(\sin A\) and \(\cos A\).

7. **Finding Angle A**:
After solving the above equation, we find that:
\[ A = 75^\circ \]

### Conclusion:
Thus, the measure of angle A is \(75^\circ\).