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 = 37°, a = 12.4, b = 8.7

Verified step by step guidance

1. Identify the given values: angle \(B = 37^\circ\), side \(a = 12.4\), and side \(b = 8.7\).
2. Use the Law of Sines to find angle \(A\): \(\frac{a}{\sin A} = \frac{b}{\sin B}\).
3. Rearrange the equation to solve for \(\sin A\): \(\sin A = \frac{a \cdot \sin B}{b}\).
4. Calculate \(\sin A\) and determine if \(A\) is possible (i.e., \(0 \leq \sin A \leq 1\)).
5. If \(A\) is possible, use \(A + B + C = 180^\circ\) to find angle \(C\), and then use the Law of Sines to find side \(c\).

Key Concepts

Law of Sines

The Law of Sines is a fundamental principle in trigonometry that relates the ratios of the lengths of sides of a triangle to the sines of its angles. It states that for any triangle, the ratio of a side length to the sine of its opposite angle is constant. This law is particularly useful for solving triangles when given two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).

Ambiguous Case of the Law of Sines

The ambiguous case occurs when using the Law of Sines with the SSA condition, where two sides and a non-included angle are known. This situation can lead to zero, one, or two possible triangles, depending on the given measurements. Understanding this concept is crucial for determining whether one or two triangles can be formed and for accurately solving the triangle.

Triangle Sum Theorem

The Triangle Sum Theorem states that the sum of the interior angles of a triangle is always 180 degrees. This theorem is essential for finding unknown angles in a triangle once other angles are known. In the context of solving triangles, it helps ensure that the calculated angles are valid and aids in determining the measures of all angles when some are already provided.