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For acute angles A and B and A + 2B and 2A + B are acute if $\tan (A + 2B) = \sqrt{3}$ and $\sin (2A + B) = \frac{1}{\sqrt{2}}$ then find the measures of angles A and B.
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Sol. $\tan(A + 2B) = \sqrt{3} \Rightarrow A + 2B = 60^\circ$ (I) (1/2)
$ \sin(2A + B) = \frac{1}{\sqrt{2}} \Rightarrow 2A + B = 45^\circ$ (II) (1/2)
On solving above equations, $A = 10^\circ$, $B = 25^\circ$ (III) (1/2+1/2)
$ \sin(2A + B) = \frac{1}{\sqrt{2}} \Rightarrow 2A + B = 45^\circ$ (II) (1/2)
On solving above equations, $A = 10^\circ$, $B = 25^\circ$ (III) (1/2+1/2)