Solve each triangle. See Examples 2 and 3.

a = 42.9 m, b = 37.6 m, c = 62.7 m

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Identify the given sides of the triangle: \(a = 42.9\) m, \(b = 37.6\) m, and \(c = 62.7\) m. 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 \(C\) opposite side \(c\), apply the formula: \[\cos C = \frac{a^2 + b^2 - c^2}{2ab}\]

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

Next, use the Law of Cosines again to find another angle, such as angle \(A\) opposite side \(a\): \[\cos A = \frac{b^2 + c^2 - a^2}{2bc}\] and then \[A = \arccos\left(\frac{b^2 + c^2 - a^2}{2bc}\right)\]

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

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Triangle Side Lengths and Classification

Understanding the given side lengths helps classify the triangle (scalene, isosceles, or equilateral) and determines which methods to use for solving it. Here, all sides are different, indicating a scalene triangle, which requires specific laws to find angles.

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The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is essential for finding unknown angles when all three sides are known, using the formula c² = a² + b² - 2ab cos(C).

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The sum of the interior angles in any triangle is always 180 degrees. After finding one or two angles using the Law of Cosines, this property allows calculation of the remaining angle to fully solve the triangle.

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