Solve each triangle. See Examples 2 and 3.


a = 3.0 ft, b = 5.0 ft, c = 6.0 ft

Verified step by step guidance

1. Identify the given sides of the triangle: \(a = 3.0\) ft, \(b = 5.0\) ft, and \(c = 6.0\) ft. Since all three sides are known, this is a side-side-side (SSS) triangle problem.
2. Use the Law of Cosines to find one of the angles. For example, to find angle \(C\) opposite side \(c\), use the formula: \[\cos C = \frac{a^2 + b^2 - c^2}{2ab}\]
3. Calculate \(\cos C\) by substituting the known side lengths into the formula, then find angle \(C\) by taking the inverse cosine (arccos) of that value: \[C = \cos^{-1}\left(\frac{a^2 + b^2 - c^2}{2ab}\right)\]
4. Repeat the Law of Cosines to find another angle, for example angle \(B\) opposite side \(b\(, using: \[\cos B = \frac{a^2 + c^2 - b^2}{2ac}\] and then find \)B = \cos^{-1}(\text{value})\).
5. Find the third angle \(A\) using the fact that the sum of angles in a triangle is \(180^\circ\): \[A = 180^\circ - B - C\].

Key Concepts

Law of Cosines

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is especially useful for solving triangles when all three sides are known, allowing calculation of each angle using the formula: c² = a² + b² - 2ab·cos(C).

Triangle Classification and Properties

Understanding the types of triangles (acute, obtuse, or right) based on side lengths helps in interpreting results. The triangle inequality theorem ensures the given sides can form a triangle, and recognizing side relationships aids in predicting angle sizes.

Angle Sum Property of Triangles

The sum of the interior angles in any triangle is always 180 degrees. After finding two angles using the Law of Cosines, the third angle can be found by subtracting their sum from 180°, completing the solution of the triangle.