Solve each triangle ABC that exists.
A = 42.5°, a = 15.6 ft, b = 8.14 ft

Verified step by step guidance

1. Step 1: Identify the given information: angle \( A = 42.5^\circ \), side \( a = 15.6 \) ft, and side \( b = 8.14 \) ft.
2. Step 2: Use the Law of Sines to find angle \( B \). The Law of Sines states \( \frac{a}{\sin A} = \frac{b}{\sin B} \). Substitute the known values into the equation.
3. Step 3: Solve for \( \sin B \) by rearranging the equation: \( \sin B = \frac{b \cdot \sin A}{a} \). Calculate \( \sin B \) using the given values.
4. Step 4: Determine angle \( B \) by taking the inverse sine (\( \sin^{-1} \)) of the value obtained for \( \sin B \).
5. Step 5: Find angle \( C \) using the fact that the sum of angles in a triangle is \( 180^\circ \). Calculate \( C = 180^\circ - A - B \).

Key Concepts

Law of Sines

The Law of Sines states that the ratios of the lengths of the sides of a triangle to the sines of their opposite angles are equal. This relationship is crucial for solving triangles when given two angles and one side or two sides and a non-included angle. In this case, it will help find the unknown angles and side lengths of triangle ABC.

Triangle Sum Theorem

The Triangle Sum Theorem asserts that the sum of the interior angles of a triangle is always 180 degrees. This theorem is essential for determining the third angle of triangle ABC once two angles are known, allowing for the complete solution of the triangle.

Ambiguous Case of the Law of Sines

The Ambiguous Case occurs when using the Law of Sines with two sides and a non-included angle, potentially leading to zero, one, or two possible triangles. Understanding this concept is vital for correctly interpreting the results when solving triangle ABC, ensuring all possible configurations are considered.