Trigonometry & Inverse Trigonometry
System of arcsin Equations — Finding γ
nta_pyq_2024_jan
Grade 12
Question:
For $\alpha,\beta,\gamma\neq0$. If $\sin^{-1}\alpha+\sin^{-1}\beta+\sin^{-1}\gamma=\pi$ and $(\alpha+\beta+\gamma)(\alpha-\gamma+\beta)=3\alpha\beta$, then $\gamma$ equals
$\dfrac{\sqrt{3}}{2}$
$\dfrac{1}{\sqrt{2}}$
$\dfrac{\sqrt{3}-1}{2\sqrt{2}}$
$\sqrt{3}$
Step-by-Step Solution
Key Concept: Let $A=\sin^{-1}\alpha,B=\sin^{-1}\beta,C=\sin^{-1}\gamma$ so $A+B+C=\pi$. Expand the second condition to get $\frac{\alpha^2+\beta^2-\gamma^2}{2\alpha\beta}=\frac{1}{2}$, which means $\cos C=\frac{1}{2}$, giving $C=\pi/3$ and $\gamma=\sin(\pi/3)=\sqrt3/2$.
$\cos C=\frac{\alpha^2+\beta^2-\gamma^2}{2\alpha\beta}=\frac{1}{2}\Rightarrow C=\pi/3\Rightarrow\gamma=\sin(\pi/3)=\frac{\sqrt3}{2}$.
Correct Answer: 1