If α and β are positive acute angles, sin(4α - β) = 1 and cos(2α + β) = $$\frac{1}{2}{\text{,}}$$ then the value of sin(α + 2β) is?

Solution(By Examveda Team)

$$\eqalign{ & {\text{sin}}\left( {4\alpha - \beta } \right) = 1 = \sin {90^ \circ } \cr & 4\alpha - \beta = {90^ \circ }{\text{ }} \cr & {\text{cos}}\left( {2\alpha + \beta } \right) = \frac{1}{2} = \cos {60^ \circ } \cr & 2\alpha + \beta = {60^ \circ } \cr & {\text{Adding }}6\alpha = {150^ \circ } \cr & \alpha = {25^ \circ } \cr & \beta = {10^ \circ } \cr & \Rightarrow {\text{sin}}\left( {\alpha + 2\beta } \right) \cr & \Rightarrow {\text{sin}}\left( {{{25}^ \circ } + 2 \times {{10}^ \circ }} \right) \cr & \Rightarrow {\text{sin}}{45^ \circ } \cr & \Rightarrow \frac{1}{{\sqrt 2 }} \cr} $$