If the angle of triangle A, B and C are in AP and b : a = √3 : 1, then what is the value of A?

Concepts:

Sine law ⇔ \(\frac{{\rm{a}}}{{\sin {\rm{A}}}} = {\rm{\;}}\frac{{\rm{b}}}{{\sin {\rm{B}}}} = {\rm{\;}}\frac{{\rm{c}}}{{\sin {\rm{C}}}}\)

Where a, b and c are sides and A, B and C are angles.

Here; Side a faces angle A, side b faces angle B and side c faces angle C

Calculation:

Here, A, B and C are in AP

So, 2B = A + C

And sum of angles of triangle, A + B + C = 180°

⇒ 2B + B = 180 

⇒ 3B = 180 

⇒ B = 60° 

Now, by sine rule,

\(\frac{{\rm{a}}}{{\sin {\rm{A}}}} = {\rm{\;}}\frac{{\rm{b}}}{{\sin {\rm{B}}}} \)

\(\rm ⇒ \frac{b}{a}=\frac{\sin B}{\sin A}\)

\(\rm ⇒ \frac{√3 }{1}=\frac{\sin 60^o}{\sin A}\)

⇒ sin A = (√3/2) × (1/√3)

⇒ sin A = 1/2

⇒ A = 30° 

Hence, option (1) is correct.