For , , 0, if ^-1 + ^-1 + ^-1 = and ( + + )( - + )=3 , then equals

Question

For $\alpha ,\beta ,\gamma 
e 0$, 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

Let $$\sin ^{-1} \alpha=A, \sin ^{-1} \beta=B, \sin ^{-1} \gamma=C$$

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

Solutions · 1