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The angles $\alpha, \beta, \gamma$ of a triangle satisfy the equations $2 \sin \alpha+3 \cos \beta=3 \sqrt{2}$ and $3 \sin \beta+2 \cos \alpha=1$. Then, angle $\gamma$ equals
$150^{\circ}$
$120^{\circ}$
$60^{\circ}$
$30^{\circ}$
Solution
(d)
We have,
$\alpha, \beta, \gamma$ are angle of triangle.
$\therefore \quad \alpha+\beta+\gamma=180^{\circ}$
Given, $\quad 2 \sin \alpha+3 \cos \beta=3 \sqrt{2}$
and $\quad 3 \sin \beta+2 \cos \alpha=1$
On squaring and adding Eqs. $(i)$ and $(ii)$
we get
$4 \sin ^2 \alpha+9 \cos ^2 \beta+12 \sin \alpha \cos \beta$
$+9 \sin ^2 \beta+4 \cos ^2 \alpha$
$+12 \cos \alpha \sin \beta=18+1$
$\Rightarrow 4\left(\sin ^2 \alpha+\cos ^2 \alpha\right)+9\left(\cos ^2 \beta+\sin ^2 \beta\right)$
$+12(\sin \alpha \cos \beta+\cos \alpha \sin \beta)=19$
$\Rightarrow \quad 4+9+12 \sin (\alpha+\beta)=19$
$\Rightarrow \quad 12 \sin (\alpha+\beta)=19-9-4$
$\Rightarrow \sin (\alpha+\beta) =\frac{1}{2}$
$\Rightarrow \alpha+\beta =150^{\circ}$
$\therefore \gamma =30^{\circ}$