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# Solving Triangles

Trigonometric problem solving culminates in this chapter. Leave no side and no angle unmeasured!

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 \(ABC\) 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.

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