In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively.
a = 1.4, b = 2.9, A = 142°

Verified step by step guidance

1. Identify the given elements: side \(a = 1.4\), side \(b = 2.9\), and angle \(A = 142^\circ\). Note that angle \(A\) is opposite side \(a\).
2. Check the type of triangle and the possibility of solutions using the Law of Sines. The Law of Sines states: \(\frac{a}{\sin A} = \frac{b}{\sin B}\).
3. Calculate \(\sin B\) using the formula \(\sin B = \frac{b \sin A}{a}\). Substitute the known values: \(\sin B = \frac{2.9 \times \sin 142^\circ}{1.4}\).
4. Analyze the value of \(\sin B\) to determine the number of possible triangles: if \(\sin B > 1\), no triangle exists; if \(\sin B = 1\), one right triangle exists; if \(0 < \sin B < 1\), two possible triangles may exist (ambiguous case).
5. If one or two triangles exist, find angle \(B\) by taking \(\sin^{-1}(\sin B)\), then find angle \(C\) using \(C = 180^\circ - A - B\). Finally, use the Law of Sines again to find side \(c\) with \(c = \frac{a \sin C}{\sin A}\).

Key Concepts

Law of Sines

The Law of Sines relates the ratios of the lengths of sides of a triangle to the sines of their opposite angles. It is essential for solving triangles when two sides and a non-included angle (SSA) are given, allowing calculation of unknown angles or sides.

Ambiguous Case of SSA Triangles

The SSA configuration can produce zero, one, or two possible triangles depending on the given measurements. Understanding this ambiguity is crucial to determine how many triangles satisfy the given conditions before solving.

Angle and Side Measurement Constraints

Triangle angle sums must equal 180°, and side lengths must satisfy the triangle inequality. These constraints help verify the validity of solutions and ensure that calculated angles and sides form a possible triangle.