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Solving Triangles

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

Extended Sine Rule

         

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.

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