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: Angles in AP:
Let the angles be $60^{\circ}-d,\ 60^{\circ},\ 60^{\circ}+d$.

Step 2: Use the ratio of sides:
$c:b=\sqrt{2}:\sqrt3$. By the sine rule $\sin C=\dfrac{c}{b}\sin60^{\circ}=\sqrt{\dfrac23}\cdot\dfrac{\sqrt3}{2}=\dfrac{\sqrt2}{2}$.

Step 3: Finish:
$C=45^{\circ}$, $A=180^{\circ}-105^{\circ}=75^{\circ}=\dfrac{5\pi}{12}$. Option A.

Final Answer:
B = 60 and C = 45 degrees, so A = 75 degrees. \[ \boxed{\text{(A) }\dfrac{5\pi}{12}} \]