Question

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

Solution

## Given, $$\displaystyle \sin A+\cos 2A=\frac { 1 }{ 2 } ,\cos A+\sin 2A=\frac { 1 }{ 3 }$$Squaring and adding $$\sin ^{ 2 }{ A } +\sin ^{ 2 }{ 2A } +2\sin { A } \cos { 2A } +\cos ^{ 2 }{ A } +\cos ^{ 2 }{ 2A } +2\cos { A } \sin { 2A } =\frac { 1 }{ 4 } +\frac { 1 }{ 9 }$$$$\displaystyle \Rightarrow 2+2\sin { 3A=\frac { 13 }{3 6 } \Rightarrow \sin { 3A } =\frac { -59 }{ 72 } }$$Mathematics

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