In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle. B = 66°, a = 17, c = 12

Verified step by step guidance

1. Identify the given elements of the triangle: angle \(B = 66^\circ\), side \(a = 17\), and side \(c = 12\). We need to find the remaining sides and angles of the triangle.
2. Use the Law of Sines, which states \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\), to find angle \(C\) or angle \(A\). Since we know \(a\), \(c\), and \(B\), start by finding angle \(C\) using \(\frac{a}{\sin A} = \frac{c}{\sin C}\) or find \(A\) first by relating \(a\) and \(B\).
3. Calculate angle \(A\) using the Law of Sines: \(\frac{a}{\sin A} = \frac{c}{\sin C}\) can be rearranged to find \(\sin A\) if you find \(\sin C\) first, or alternatively use \(\frac{a}{\sin A} = \frac{b}{\sin B}\) if \(b\) is known. Since \(b\) is unknown, try to find \(\sin A\) by using \(\frac{a}{\sin A} = \frac{c}{\sin C}\) and express \(\sin C\) in terms of \(B\) and \(A\).
4. Check for the possibility of two triangles by considering the ambiguous case of the Law of Sines (SSA configuration). Calculate \(\sin A\) and verify if it is less than or equal to 1. If \(\sin A < 1\), then two possible angles for \(A\) exist: \(A\) and \(180^\circ - A\). This means two triangles may be possible.
5. Once angles \(A\) and \(C\) are found, use the fact that the sum of angles in a triangle is \(180^\circ\) to find the missing angle. Then use the Law of Sines again to find the missing side \(b\). Round all lengths to the nearest tenth and angles to the nearest degree as required.

Key Concepts

Law of Sines

The Law of Sines relates the ratios of sides to the sines of their opposite angles in any triangle: (a/sin A) = (b/sin B) = (c/sin C). It is essential for solving triangles when given two angles and a side or two sides and a non-included angle, as in this problem.

Ambiguous Case of the Law of Sines (SSA Condition)

When two sides and a non-included angle (SSA) are given, there can be zero, one, or two possible triangles. This ambiguity arises because the given angle and side lengths may produce no triangle, a unique triangle, or two distinct triangles, requiring careful analysis.

Triangle Angle Sum Property

The sum of the interior angles in any triangle is always 180°. After finding one unknown angle using the Law of Sines, this property helps determine the remaining angle, completing the triangle's angle measures.