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 = 10, b = 30, A = 150°

Verified step by step guidance

1. Identify the given elements: side \(a = 10\), side \(b = 30\), and angle \(A = 150^\circ\). Note that angle \(A\) is opposite side \(a\).
2. Use the Law of Sines to find the possible value(s) of angle \(B\). The Law of Sines states: \(\frac{a}{\sin A} = \frac{b}{\sin B}\). Substitute the known values to get \(\frac{10}{\sin 150^\circ} = \frac{30}{\sin B}\).
3. Solve for \(\sin B\) by rearranging the equation: \(\sin B = \frac{30 \times \sin 150^\circ}{10}\). Calculate the right side to find the value of \(\sin B\) (do not compute the final value here).
4. Determine the number of possible triangles by analyzing the value of \(\sin B\): if \(\sin B > 1\), no triangle exists; if \(\sin B = 1\), one right triangle exists; if \(0 < \sin B < 1\), there are two possible angles for \(B\) (one acute and one obtuse), so two triangles may exist.
5. For each possible angle \(B\), find angle \(C\) using the triangle angle sum: \(C = 180^\circ - A - B\). Then use the Law of Sines again to find side \(c\): \(\frac{c}{\sin C} = \frac{a}{\sin A}\). Round all sides to the nearest tenth and angles to the nearest degree.

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. This ambiguity arises because the given angle and side may correspond to different triangle configurations, requiring careful analysis to determine the number of solutions.

Triangle Angle Sum Property

The sum of the interior angles of any triangle is always 180°. This property is used to find the third angle once two angles are known, ensuring the triangle's validity and helping to complete the solution after applying the Law of Sines.