Solve each triangle ABC that exists.


B = 39.68°, a = 29.81 m, b = 23.76 m

Verified step by step guidance

1. Identify the given elements of the triangle: angle \(B = 39.68^\circ\), side \(a = 29.81\) m (opposite angle \(A\)), and side \(b = 23.76\) m (opposite angle \(B\)). We need to find the remaining parts of the triangle: angle \(A\), angle \(C\), and side \(c\).
2. Use the Law of Sines to find angle \(A\). The Law of Sines states: \(\frac{a}{\sin A} = \frac{b}{\sin B}\). Rearranged to solve for \(\sin A\), it becomes \(\sin A = \frac{a \sin B}{b}\).
3. Calculate \(\sin A\) using the values of \(a\), \(b\), and \(B\). Then find angle \(A\) by taking the inverse sine (arcsin) of \(\sin A\). Remember to consider the possible ambiguous case in the Law of Sines where two different angles could satisfy the sine value.
4. Once angle \(A\) is found, calculate angle \(C\) using the fact that the sum of angles in a triangle is \(180^\circ\): \(C = 180^\circ - A - B\).
5. Finally, use the Law of Sines again to find side \(c\) by applying \(\frac{c}{\sin C} = \frac{a}{\sin A}\), and solve for \(c\).

Key Concepts

Law of Sines

The Law of Sines relates the sides and angles of a triangle through the ratio a/sin(A) = b/sin(B) = c/sin(C). It is essential for solving triangles when two sides and an angle are known, allowing calculation of unknown angles or sides.

Triangle Angle Sum Property

The sum of the interior angles in any triangle is always 180°. Knowing one or two angles allows determination of the remaining angle, which is crucial for completing the triangle solution.

Ambiguous Case of the Law of Sines (SSA Condition)

When two sides and a non-included angle are given (SSA), there may be zero, one, or two possible triangles. Understanding this ambiguity helps determine all valid solutions or recognize when no triangle exists.