If $$\alpha$$, $$\beta$$ ,$$ \gamma$$ are acute angles, determine the value of $$\alpha -\beta$$ such that $$\sin \alpha +\sin \gamma = \sin \beta$$ , $$\cos \alpha -\cos \gamma = \cos \beta $$

Question

Uniteasy Admin · Accepted Answer

Express $$\gamma $$ in terms of $$\alpha$$ and $$\beta $$ 
$$\sin \gamma = \sin \beta - \sin \alpha$$
$$\cos \gamma =\cos \alpha - \cos \beta$$
Apply Pythagorean Identity for $$\gamma $$
$$\sin^2\gamma+\cos^2\gamma = ( \sin \beta - \sin \alpha)^2+(\cos \alpha - \cos \beta)^2 = 1
$$
$$ 2- 2( \cos \alpha\cos \beta +\sin \alpha \sin \beta) = 1 $$
$$\cos(\alpha -\beta ) = \dfrac{1}{2} $$
Since $$\alpha$$, $$\beta$$ ,$$ \gamma$$are acute angles,
$$\sin \beta - \sin \alpha = \sin \gamma > 0$$
$$\alpha - \beta < 0 $$
$$	herefore \alpha -\beta = - \dfrac{\pi}{3}  $$

Question

Answer