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Solving Triangles

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

Sine Rule - Ambiguous Case

         

How many distinct triangles are there such that \[a=6, c=15, \angle A=30^\circ?\]

Note: \(a, b\) and \(c\) are the lengths of the sides opposite the vertices \(A,\) \(B\) and \(C,\) respectively.

How many distinct triangles are there such that \[a=20, c=16, \angle A=30^\circ?\]

Note: \(a, b\) and \(c\) are the lengths of the sides opposite the vertices \(A,\) \(B\) and \(C,\) respectively.

How many distinct triangles are there such that \[\angle A=30^\circ, a=9, b=9\sqrt{3} ?\]

Note: \(a, b\) and \(c\) are the lengths of the sides opposite the vertices \(A,\) \(B\) and \(C,\) respectively.

In triangle \(ABC\), \(a=13\sqrt{3}\), \(b=13\), \(\angle B=30^{\circ}\), and \(\angle C\) is acute. What is the value of \(\angle C\) (in degrees)?

Details and assumptions

\(a\), \(b\) and \(c\) are the lengths of the sides opposite to the vertices \(A\), \(B\) and \(C\), respectively. An acute angle is an angle strictly less than \(90^\circ\).

How many distinct triangles are there such that \[a=35, b=56, \angle A=30^\circ?\]

Note: \(a, b\) and \(c\) are the lengths of the sides opposite the vertices \(A,\) \(B\) and \(C,\) respectively.

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