First slide

Compound angles

Question

If the angles $A, B, C$ of a triangle are in $A . P .$ such that $\sin (2 A + B) = 1 / 2$ then $\sin (B + 2 C) =$

Moderate

A

$- \frac{1}{2}$
B

$\frac{1}{2}$
C

$\frac{\sqrt{3}}{2}$
D

$\frac{1}{\sqrt{2}}$

Solution

$B = 60^{\circ}, \sin (2 A + B) = 1 / 2 \Rightarrow 2 A + B = 150^{\circ}$
$\Rightarrow A = 45^{\circ}, C = 75^{\circ}$
so that $B + 2 C = 210^{\circ} = 180^{\circ} + 30^{\circ}$
$\Rightarrow \sin (B + 2 C) = - \sin 30^{\circ} = - 1 / 2$

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