Question

The angles of \(△ABC\) are in A.P. and \(b:c = \sqrt{3}:\sqrt{2}\) then \(\angle A =\)

• A. \(30^{\circ}\)
• B. \(90^{\circ}\)
• C. \(105^{\circ}\)
• D. \(75^{\circ}\)

Hint:

Angles in A.P. force B = 60 degrees; the sine rule then gives C and A.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
If the three angles are in A.P., the middle angle is the mean of the other two. Their sum is \(180^\circ\), so \(3B=180^\circ\) and \(B=60^\circ\).

Step 2: Use the sine rule
\[ \frac{b}{c}=\frac{\sin B}{\sin C}\Rightarrow\frac{\sqrt3}{\sqrt2}=\frac{\sin60^\circ}{\sin C} \]
\[ \sin C=\frac{\sqrt3}{2}\cdot\frac{\sqrt2}{\sqrt3}=\frac{\sqrt2}{2}\]

Step 3: Find C
\(C=45^\circ\) or \(135^\circ\). The value \(135^\circ\) cannot hold because \(B+C=195^\circ\) exceeds \(180^\circ\). So \(C=45^\circ\).

Step 4: Find A
\[ A=180^\circ-60^\circ-45^\circ=75^\circ \]
Also the A.P. check: 45, 60, 75 is an A.P. The answer is option (D).

Final Answer:
The angles are 45, 60 and 75 degrees, so angle A is 75 degrees, option (D). \[ \boxed{75^{\circ}} \]