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

If triangle \(ABC\) has side lengths \(a=7\), \(b=12\) and \(c=11\), the value of \( \cos A\) can be expressed as \(\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. What is the value of \(p+q\)?

**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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Let \(ABC\) be a triangle such that \(\angle A = 30^{\circ} \), \( a = 9 \sqrt{3} \) and \( c = 18 \sqrt{3} \). 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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If triangle \(ABC\) has \( \angle A = 45^{\circ}\), \(b=28\) and \(c=\sqrt{6}+14\sqrt{2}\), what is the value of \(a^2\)?

**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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