Question

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 \( \gamma \) is equal to

• A. \( \frac{\sqrt{3}}{2} \)
• B. \( \frac{1}{\sqrt{2}} \)
• C. \( \frac{\sqrt{3} - 1}{2\sqrt{2}} \)
• D. \( \sqrt{3} \)

Answer 1

The Correct Option is A

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

$A + B + C = \pi$

$(\alpha + \beta)^2 - \gamma^2 = 3 \alpha \beta$

$\alpha^2 + \beta^2 - \gamma^2 = \alpha \beta$

$\frac{\alpha^2 + \beta^2 - \gamma^2}{2 \alpha \beta} = \frac{1}{2}$

$\Rightarrow \cos C = \frac{1}{2}$

$\sin C = \gamma$

$\cos C = \sqrt{1 - \gamma^2} = \frac{1}{2}$

$\gamma = \frac{\sqrt{3}}{2}$