Solving Triangles
In triangle \(ABC\), let \(a, b\) and \(c\) be the lengths of the legs of a triangle opposite vertices \(A, B, C \) respectively. Let \( \alpha, \beta, \gamma \) be the values of the respective angles. What is the value of
\[ a ( \sin \beta - \sin \gamma) + b ( \sin \gamma - \sin \alpha ) + c ( \sin \alpha - \sin \beta )? \]
Triangle \(ABC\) is inscribed in a circle \(\Gamma\). If \(\angle A = 60^{\circ} \) and the length of the side opposite to vertex \(A\) is \(49 \sqrt{3}\), what is the radius of \(\Gamma\)?
In triangle \(ABC\), \( \angle ABC = \frac{ \pi}{2} \). If \( AC = 12 \), what is the radius of the circumcicle of \( ABC \)?
Triangle \(ABC\) has \(\angle A=150^{\circ}\) and \(\overline{BC}=7\). If the area of the circumscribed circle of \(ABC\) is \(a\pi\), what is the value of \(a\)?
Triangle \(ABC\) satisfies \(2\sin B=\sin A+\sin C\). If \(a+c=30\), what is the value of \(b\)?
Details and assumptions
\(a\), \(b\) and \(c\) are the lengths of the sides opposite to the vertices \(A\), \(B\) and \(C\), respectively.