If α + β − γ = π, Find λ
Question:
\[
\alpha+\beta-\gamma=\pi
\]
and
\[
\sin^2\alpha+\sin^2\beta-\sin^2\gamma
=
\lambda \sin\alpha\sin\beta\cos\gamma
\]
Find \(\lambda\).
Solution
\[ \gamma=\alpha+\beta-\pi \]
\[ \cos\gamma = -\cos(\alpha+\beta) \]
\[ = \sin\alpha\sin\beta-\cos\alpha\cos\beta \]
\[ \sin^2\gamma = \sin^2(\alpha+\beta) \]
\[ = (\sin\alpha\cos\beta+\cos\alpha\sin\beta)^2 \]
\[ = \sin^2\alpha\cos^2\beta + \cos^2\alpha\sin^2\beta + 2\sin\alpha\sin\beta\cos\alpha\cos\beta \]
\[ \sin^2\alpha+\sin^2\beta-\sin^2\gamma \]
\[ = 2\sin\alpha\sin\beta (\sin\alpha\sin\beta-\cos\alpha\cos\beta) \]
\[ = 2\sin\alpha\sin\beta\cos\gamma \]
\[ \therefore \lambda=2 \]
\[ \boxed{2} \]