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Properties of Argument

Let α, β, ...

Question

Let $α, β, γ, δ ϵ (0, π)$ be such that $2 cos α + 6 cos β + 7 cos γ + 9 cos δ = 0$ and $2 sin α - 6 sin β + 7 sin γ - 9 sin δ = 0,$ then the value of $λ$ for which $λ cos (α + δ) = 7 cos (β + γ)$ is

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Solution

Given $α, β, γ, δ ϵ (0, π)$
$2 cos α + 6 cos β + 7 cos γ + 9 cos δ = 0$
$2 sin α - 6 sin β + 7 sin γ - 9 sin δ = 0,$

Now,
$2 cos α + 6 cos β + 7 cos γ + 9 cos δ = 0$ [given]
$\Rightarrow 2 cos α + 9 cos δ = - (6 cos β + 7 cos γ)$
$\Rightarrow (2 cos α + 9 cos δ)^{2} = (- (6 cos β + 7 cos γ))^{2}$ [Squaring on both sides]
$\Rightarrow 4 {cos}^{2} α + 81 {cos}^{2} δ + 36 cos α cos δ = 36 {cos}^{2} β + 49 {cos}^{2} γ + 84 cos β cos γ$ $\dots (i)$

and
$2 sin α - 6 sin β + 7 sin γ - 9 sin δ = 0$ [given]
$\Rightarrow 2 sin α - 9 sin δ = 6 sin β - 7 sin γ$
$\Rightarrow (2 s i n α - 9 sin δ)^{2} = (6 sin β - 7 sin γ)^{2}$ [Squaring on both sides]
$\Rightarrow 4 {sin}^{2} α + 81 {sin}^{2} δ - 36 sin α sin δ = 36 {sin}^{2} β + 49 {sin}^{2} γ - 84 sin β sin γ$ $\dots (i i)$

Add $(i), (i i)$
$\Rightarrow 4 {cos}^{2} α + 81 {cos}^{2} δ + 36 cos α cos δ + 4 {sin}^{2} α + 81 {sin}^{2} δ - 36 sin α sin δ$
$= 36 {cos}^{2} β + 49 {cos}^{2} γ + 84 cos β cos γ + 36 {sin}^{2} β + 49 {sin}^{2} γ - 84 sin β sin γ$
$\Rightarrow 4 {cos}^{2} α + 4 {sin}^{2} α + 81 {sin}^{2} δ + 81 {cos}^{2} δ + 36 cos α cos δ - 36 sin α sin δ$
$= 36 {cos}^{2} β + 36 {sin}^{2} β + 49 {cos}^{2} γ + 49 {cos}^{2} γ + 84 cos β cos γ - 84 sin β sin γ$
$\Rightarrow 4 [{cos}^{2} α + {sin}^{2} α] + 81 [{sin}^{2} δ + {cos}^{2} δ] + 36 [cos α cos δ - sin α sin δ]$
$= 36 [{cos}^{2} β + {sin}^{2} β] + 49 [{cos}^{2} γ + {cos}^{2} γ] + 84 [cos β cos γ - sin β sin γ]$
$\Rightarrow 4 (1) + 81 (1) + 36 (cos (α + δ)) = 36 (1) + 49 (1) + 84 (cos (β + γ))$
$\Rightarrow 85 + 36 (cos (α + δ)) = 85 + 84 (cos (β + γ))$
$\Rightarrow 36 (cos (α + δ)) = 84 (cos (β + γ))$
$\Rightarrow 3 (cos (α + δ)) = 7 (cos (β + γ))$

by comparing with $λ cos (α + δ) = 7 cos (β + γ)$

we get $λ = 3$

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