Solve each triangle. See Examples 2 and 3.


B = 74.8°, a = 8.92 in., c = 6.43 in.

Verified step by step guidance

1. Identify the given elements of the triangle: angle \(B = 74.8^\circ\), side \(a = 8.92\) inches (opposite angle \(A\)), and side \(c = 6.43\) inches (opposite angle \(C\)). We need to find the remaining angles \(A\) and \(C\), and side \(b\).
2. Use the Law of Sines, which states: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). Since we know \(a\), \(c\), and \(B\), we can find angle \(C\) first by setting up the ratio \(\frac{a}{\sin A} = \frac{c}{\sin C}\) or use \(\frac{c}{\sin C} = \frac{a}{\sin A}\), but since \(A\) is unknown, it's easier to use \(\frac{c}{\sin C} = \frac{a}{\sin A}\) after finding \(C\) from \(B\) and sides.
3. To find angle \(C\), use the Law of Sines ratio between sides \(a\) and \(c\) and their opposite angles \(A\) and \(C\). Since \(A\) is unknown, instead use the Law of Cosines or find angle \(C\) by first finding angle \(A\) using the Law of Sines with known \(a\), \(B\), and \(c\). Alternatively, use the Law of Sines with \(a\), \(B\), and \(c\) to find \(C\) by setting up \(\frac{a}{\sin A} = \frac{c}{\sin C}\) and expressing \(A\) in terms of \(B\) and \(C\).
4. Once you find angle \(C\), calculate angle \(A\) using the fact that the sum of angles in a triangle is \(180^\circ\): \(A = 180^\circ - B - C\).
5. Finally, use the Law of Sines again to find side \(b\) by setting \(\frac{b}{\sin B} = \frac{a}{\sin A}\) and solving for \(b\).

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 expressed as (a/sin A) = (b/sin B) = (c/sin C). This law is essential for solving triangles when given two sides and an angle not included between them, as in this problem.

Triangle Angle Sum Property

The sum of the interior angles of any triangle is always 180°. Knowing one or two angles allows you to find the remaining angle(s) by subtracting from 180°. This property helps complete the triangle once some angles are known.

Ambiguous Case of the Law of Sines (SSA)

When two sides and a non-included angle (SSA) are given, there may be zero, one, or two possible triangles. This ambiguity arises because the given data can correspond to different configurations. Recognizing and resolving this case is crucial for correctly solving the triangle.