If $sin A + cos 2 A = 1 / 2$ and $cos A + sin 2 A = 1 / 3,$ then find the value of $sin 3 A$

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Solution

Given, $sin A + cos 2 A = \frac{1}{2}, cos A + sin 2 A = \frac{1}{3}$
Squaring and adding
${sin}^{2} A + {sin}^{2} 2 A + 2 sin A cos 2 A + {cos}^{2} A + {cos}^{2} 2 A + 2 cos A sin 2 A = \frac{1}{4} + \frac{1}{9}$
$\Rightarrow 2 + 2 sin 3 A = \frac{13}{36} \Rightarrow sin 3 A = \frac{- 59}{72}$

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