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Solving Triangles

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

Cosine Rule

         

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.

In a triangle, an angle of \(60^\circ\) is formed by two sides of lengths \(4\) and \(11\). If the length of the remaining side is \(\sqrt{a}\), what is \(a\)?

If parallelogram \(ABCD\) has \(\overline{AB}=8\), \(\overline{BC}=4 \) and \(\angle ABC=60^{\circ}\), what is the square of the length of the diagonal \(\overline{BD}\)?

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.

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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