Solve each triangle ABC.
A = 68.41°, B = 54.23°, a = 12.75 ft

Verified step by step guidance

1. Step 1: Use the angle sum property of triangles to find angle C. Since the sum of angles in a triangle is 180°, calculate C = 180° - A - B.
2. Step 2: Apply the Law of Sines to find side b. The Law of Sines states that \( \frac{a}{\sin A} = \frac{b}{\sin B} \). Rearrange to solve for b: \( b = \frac{a \cdot \sin B}{\sin A} \).
3. Step 3: Use the Law of Sines again to find side c. The formula is \( \frac{a}{\sin A} = \frac{c}{\sin C} \). Rearrange to solve for c: \( c = \frac{a \cdot \sin C}{\sin A} \).
4. Step 4: Calculate the sine of each angle using a calculator or trigonometric tables: \( \sin A \), \( \sin B \), and \( \sin C \).
5. Step 5: Substitute the known values into the equations from Steps 2 and 3 to find the lengths of sides b and c.

Key Concepts

Law of Sines

The Law of Sines states that the ratios of the lengths of sides of a triangle to the sines of their opposite angles are equal. This principle is crucial for solving triangles when given two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA). It allows for the calculation of unknown side lengths and angles, facilitating the solution of triangle ABC.

Angle Sum Property

The Angle Sum Property of a triangle asserts that the sum of the interior angles in any triangle is always 180 degrees. This property is essential for finding the third angle when two angles are known, which is necessary for applying the Law of Sines or the Law of Cosines effectively in triangle ABC.

Trigonometric Ratios

Trigonometric ratios relate the angles of a triangle to the lengths of its sides. In the context of solving triangles, these ratios (sine, cosine, and tangent) are used to find unknown side lengths and angles. Understanding these ratios is fundamental for applying the Law of Sines and the Law of Cosines in the solution process.