Solve each triangle ABC that exists.


B = 72.2°, b = 78.3 m, c = 145 m

Verified step by step guidance

1. Step 1: Use the Law of Sines to find angle C. The Law of Sines states \( \frac{b}{\sin B} = \frac{c}{\sin C} \). Substitute the known values: \( \frac{78.3}{\sin 72.2^\circ} = \frac{145}{\sin C} \).
2. Step 2: Solve for \( \sin C \) by rearranging the equation: \( \sin C = \frac{145 \cdot \sin 72.2^\circ}{78.3} \).
3. Step 3: Calculate angle C using the inverse sine function: \( C = \sin^{-1}(\text{value from Step 2}) \).
4. Step 4: Find angle A using the angle sum property of triangles: \( A = 180^\circ - B - C \).
5. Step 5: Use the Law of Sines again to find side a: \( \frac{a}{\sin A} = \frac{b}{\sin B} \). Rearrange to solve for a: \( a = \frac{b \cdot \sin A}{\sin B} \).

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 is expressed as a/b = sin(A)/sin(B) = c/sin(C). It is particularly useful for solving triangles when two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA) are known.

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 unknown angles in a triangle when two angles are known, allowing for the calculation of the third angle by subtracting the sum of the known angles from 180 degrees.

Ambiguous Case of SSA

The Ambiguous Case of SSA occurs when two sides and a non-included angle are known, which can lead to zero, one, or two possible triangles. This situation requires careful analysis using the Law of Sines to determine the number of valid triangles that can be formed, as well as their respective dimensions.