Jee Main 2024 31 01 2024 Shift1 - Question15
Question 15
For $\alpha, \beta \gamma>0$, if $\sin ^{-1} \alpha+\sin ^{-1} \beta+\sin ^{-1} \gamma=\pi$ and $(\alpha$ $+\beta+\gamma)(\alpha-\gamma+\beta)=3 \alpha \beta$, then $\gamma$ is
(1) $-\left(\frac{\sqrt{3}-1}{2 \sqrt{2}}\right)$
(2) $\frac{-1}{\sqrt{2}}$
(3) $-\sqrt{3}$
(4) $\frac{\sqrt{3}}{2}$
Show Answer
Answer: (4)
Solution:
Let $\sin A=\alpha$
$$ \begin{aligned} & \sin B=\beta \\ & \sin C=\gamma \\ & A+B+C=\pi \\ & \Rightarrow(\alpha+\beta)^{2}-\gamma^{2}=3 \alpha \beta \\ & \Rightarrow \alpha^{2}+\beta^{2}-\alpha \beta=\gamma^{2} \\ & \Rightarrow \alpha^{2}+\beta^{2}-\gamma^{2}=\alpha \beta \\ & \Rightarrow \frac{\alpha^{2}+\beta^{2}-\gamma^{2}}{2 \alpha \beta}=\frac{1}{2} \\ & \Rightarrow \cos C=\frac{1}{2} \Rightarrow C=60^{\circ} \\ & \Rightarrow \sin C=\frac{\sqrt{3}}{2}=\gamma \end{aligned} $$
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