Solve each triangle. See Examples 2 and 3.

a = 965 ft, b = 876 ft, c = 1240 ft

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Identify the given sides of the triangle: \(a = 965\) ft, \(b = 876\) ft, and \(c = 1240\) ft. Since all three sides are known, this is a side-side-side (SSS) triangle problem.

Use the Law of Cosines to find one of the angles. For example, to find angle \(A\) opposite side \(a\), use the formula: \[\cos A = \frac{b^2 + c^2 - a^2}{2bc}\]

Calculate \(\cos A\) using the given side lengths, then find angle \(A\) by taking the inverse cosine (arccos) of that value: \[A = \cos^{-1}\left(\frac{b^2 + c^2 - a^2}{2bc}\right)\]

Repeat the Law of Cosines to find another angle, for example angle \(B\) opposite side \(b\): \[\cos B = \frac{a^2 + c^2 - b^2}{2ac}\] Then find \(B\) by taking the inverse cosine: \[B = \cos^{-1}\left(\frac{a^2 + c^2 - b^2}{2ac}\right)\]

Find the third angle \(C\) using the fact that the sum of angles in a triangle is \(180^\circ\): \[C = 180^\circ - A - B\]

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 Angle Sum Property

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 determined by subtracting the sum of the known angles from 180°, ensuring the triangle's angle measures are consistent.

Triangle Classification by Sides

Understanding the type of triangle based on side lengths (scalene, isosceles, or equilateral) helps in anticipating the nature of angles. Given three different side lengths, the triangle is scalene, meaning all angles and sides are distinct, which influences the approach to solving it.

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