Question

In triangle ABC, with usual notations, if the angles are in Arithmetic Progression and \(3c^2 = 2b^2\), then the measure of angle A is

• A. \(\frac{5π}{12}\)
• B. \(\frac{7π}{12}\)
• C. \(\frac{5π}{6}\)
• D. \(\frac{7π}{6}\)

Hint:

The middle angle of an AP of angles is 60 degrees; then use the sine rule.

Answer 1

The Correct Option is A

Step 1: Use the AP condition:
If \(A, B, C\) are in AP then \(2B=A+C\). With \(A+B+C=180^{\circ}\) we get \(3B=180^{\circ}\), so \(B=60^{\circ}\).

Step 2: Use the sine rule:
\(\dfrac{c}{b}=\dfrac{\sin C}{\sin B}\). From \(3c^2=2b^2\), \(\dfrac{c^2}{b^2}=\dfrac23\), so \(\dfrac{\sin^2C}{\sin^260^{\circ}}=\dfrac23\).

Step 3: Find C:
\(\sin^2C=\dfrac23\cdot\dfrac34=\dfrac12\), so \(\sin C=\dfrac{1}{\sqrt2}\) and \(C=45^{\circ}\) or \(135^{\circ}\). \(135^{\circ}\) is impossible since \(B+C\) would exceed \(180^{\circ}\). So \(C=45^{\circ}\).

Step 4: Find A:
\(A=180^{\circ}-60^{\circ}-45^{\circ}=75^{\circ}=\dfrac{5\pi}{12}\). Option A.

Final Answer:
Angle A is 5 pi / 12. \[ \boxed{\text{(A) }\dfrac{5\pi}{12}} \]