Solve each triangle. See Examples 2 and 3.


A = 41.4°, b = 2.78 yd, c = 3.92 yd

Verified step by step guidance

1. Step 1: Identify the type of triangle. Since we have two sides and an angle, this is an SSA (Side-Side-Angle) triangle.
2. Step 2: Use the Law of Sines to find the missing angle B. The Law of Sines states: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$.
3. Step 3: Rearrange the Law of Sines to solve for $\sin B$: $\sin B=\frac{b\cdot \sin A}{c}$. Substitute the known values: $b=2.78$, $A={41.4}^{\circ }$, and $c=3.92$.
4. Step 4: Calculate $\sin B$ using the values from Step 3. Then, use the inverse sine function to find angle B.
5. Step 5: Use the fact that the sum of angles in a triangle is 180° to find angle C: $C={180}^{\circ }-A-B$. Then, use the Law of Sines again to find the missing side a.

Key Concepts

Law of Sines

The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides and angles. This relationship is crucial for solving triangles when given two angles and one side or two sides and a non-included angle. It allows for the calculation of unknown angles and sides, facilitating the solution of various triangle types.

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 finding missing angles in a triangle when two angles are known. Understanding this concept helps in determining the third angle, which is often necessary for applying the Law of Sines or the Law of Cosines.

Side-Angle Relationships

In any triangle, the lengths of the sides are directly related to the measures of the angles opposite those sides. Specifically, the larger the angle, the longer the side opposite it. This relationship is fundamental when using the Law of Sines or the Law of Cosines, as it helps in determining which sides and angles can be calculated based on the given information.