If in a ΔABC , ∠B = π/3, then the maximum value of sinA sinC is

Q: If in a ΔABC , ∠B = π/3, then the maximum value of sinA sinC is

(A) 1/2

(B) 3/4

(C) 2/3

(D) None of these

Sol: $\large A + C = \frac{2 \pi}{3} $

$\large C = \frac{2 \pi}{3} – A $

Now , $\large sin A sin C = sin A . sin(\frac{2 \pi}{3} – A) $

$\large = \frac{\sqrt{3}}{4} sin2A + \frac{1}{2}sin^2 A $

$\large = \frac{\sqrt{3}}{4} sin2A + \frac{1}{4}(1-cos2A) $

$\large = \frac{\sqrt{3}}{4} sin2A – \frac{1}{4}cos2A + \frac{1}{4}$

$\large – \frac{1}{2} \le sinA sinC – \frac{1}{4} \le \frac{1}{2} $

$\large – \frac{1}{4} \le sinA sinC \le \frac{3}{4} $

Hence (B) is the correct answer.