For \alpha, \beta, \gamma
eq 0, if \sin ^{-1} \alpha+\sin ^{-1} \beta+\sin ^{-1} \gamma=\pi and (\alpha+\beta+\gamma)(\alpha-\gamma+\beta)=3 \alpha \beta, then \gamma equals

Question

For \alpha, \beta, \gamma 
eq 0, if \sin ^{-1} \alpha+\sin ^{-1} \beta+\sin ^{-1} \gamma=\pi and (\alpha+\beta+\gamma)(\alpha-\gamma+\beta)=3 \alpha \beta, then \gamma equals

Accepted Answer

Answer: B. \frac{\sqrt{3}}{2} Let \sin ^{-1} \alpha=A, \sin ^{-1} \beta=B, \sin ^{-1} \gamma=C \begin{aligned} & \mathrm{A}+\mathrm{B}+\mathrm{C}=\pi \ & (\alpha+\beta)^2-\gamma^2=3 \alpha \beta \ & \alpha^2+\beta^2-\gamma^2=\alpha \beta \ & \frac{\alpha^2+\beta^2-\gamma^2}{2 \alpha \beta}=\frac{1}{2} \ & \Rightarrow \cos \mathrm{C}=\frac{1}{2} \ & \sin \mathrm{C}=\gamma \ & \cos \mathrm{C}=\sqrt{1-\gamma^2}=\frac{1}{2} \ & \gamma=\frac{\sqrt{3}}{2} \end{aligned}

For $$\alpha, \beta, \gamma \neq 0$$, if $$\sin ^{-1} \alpha+\sin ^{-1} \beta+\sin ^{-1} \gamma=\pi$$ and $$(\alpha+\beta+\gamma)(\alpha-\gamma+\beta)=3 \alpha \beta$$, then $$\gamma$$ equals

Answer

Explanation

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