Question : If $\alpha$ and $\beta$ are positive acute angles, $\sin (4\alpha -\beta )=1$ and $\cos (2\alpha +\beta)=\frac{1}{2}$, then the value of $\sin (\alpha +2\beta)$ is:
Option 1: $0$
Option 2: $1$
Option 3:
$\frac{\sqrt{3}}{2}$
Option 4:
$\frac{1}{\sqrt{2}}$
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Correct Answer:
Solution : Given: $\sin (4\alpha -\beta )=1$ and $\cos (2\alpha +\beta)=\frac{1}{2}$ $\sin (4\alpha -\beta )=1$ ⇒ $\sin (4\alpha -\beta )=\sin 90^{\circ}$ ⇒ $4\alpha -\beta= 90^{\circ}$...................................... (1) Now, $\cos (2\alpha +\beta)=\frac{1}{2}$ ⇒ $\cos (2\alpha +\beta)=\cos 60^{\circ}$ ⇒ $2\alpha +\beta=60^{\circ}$....................................(2) Solving equations 1 and 2, we get: $6\alpha=150^{\circ}$ ⇒ $\alpha=25^{\circ}$ Putting the value of $\alpha$ in equation 1, we get, $4×25^{\circ} -\beta= 90^{\circ}$ ⇒ $\beta=10^{\circ}$ Thus, $\sin (\alpha +2\beta)=\sin (25^{\circ} +2×10^{\circ})$ $=\sin (25^{\circ} +20^{\circ})$ $=\sin45^{\circ}$ $=\frac{1}{\sqrt{2}}$ Hence, the correct answer is $\frac{1}{\sqrt{2}}$.
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