First slide

Multiple and sub- multiple Angles

Question

If $α, β, γ, δ$ are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity k, then the value of $4 \sin \frac{α}{2} + 3 \sin \frac{β}{2} + 2 \sin \frac{γ}{2} + \sin \frac{δ}{2}$ is equal to

Moderate

A

$2 \sqrt{1 - k}$
B

$2 \sqrt{1 + k}$
C

$\frac{\sqrt{1 + k}}{2}$
D

none of these

Solution

$Since α < β < γ < δ and \sin α = \sin β = \sin γ = \sin δ = k, we have$
$\begin{array}{l} β = π - α, γ = 2 π + α, δ = 3 π - α \\ \Rightarrow 4 \sin \frac{α}{2} + 3 \sin \frac{β}{2} + 2 \sin \frac{γ}{2} + \sin \frac{δ}{2} \\ = 4 \sin \frac{α}{2} + 3 \cos \frac{α}{2} - 2 \sin \frac{α}{2} - \cos \frac{α}{2} \\ = 2 \sin \frac{α}{2} + 2 \cos \frac{α}{2} = 2 \sqrt{1 + \sin α} = 2 \sqrt{1 + k} \end{array}$

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