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

In a triangle \(ABC\), the measures of the angles are \(\angle A=30^\circ,\) \(\angle B=80^\circ.\) Point \(M\) lies inside the triangle, such that \(\angle MAC = 10^\circ,\) \(\angle MCA = 30^\circ.\) What is the measure (in degrees) of \(\angle BMC?\)

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In a triangle \(ABC\) \(\angle A =84^\circ,\) \(\angle C=78^\circ.\) Points \(D\) and \(E\) are taken on the sides \(AB\) and \(BC,\) so that \(\angle ACD =48^\circ,\) \(\angle CAE =63^\circ.\) What is the measure (in degrees) of \(\angle CDE\) ?

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In triangle \( ABC \), \( \cos A : \cos B : \cos C = 2 : 9 : 12. \)

If \( \sin A : \sin B : \sin C = p : q : r \), where \( p \), \( q \) and \( r \) are coprime positive integers, find the value of \( pqr-(p+q+r). \)

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