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

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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