Solve each triangle. See Examples 2 and 3.

A = 41.4°, b = 2.78 yd, c = 3.92 yd

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Identify the given elements of the triangle: angle \(A = 41.4^\circ\), side \(b = 2.78\) yd, and side \(c = 3.92\) yd. We need to find the remaining sides and angles.

Use the Law of Cosines to find side \(a\) opposite angle \(A\). The formula is: \[a^2 = b^2 + c^2 - 2bc \cos A\]

Calculate \(a\) by taking the square root of the expression from the Law of Cosines: \[a = \sqrt{b^2 + c^2 - 2bc \cos A}\]

Next, use the Law of Sines to find another angle, for example angle \(B\). The Law of Sines states: \[\frac{\sin A}{a} = \frac{\sin B}{b}\] Rearranged to solve for \(B\): \[\sin B = \frac{b \sin A}{a}\]

Find angle \(B\) by taking the inverse sine (arcsin) of the value calculated in the previous step. Then, find angle \(C\) by using the fact that the sum of angles in a triangle is \(180^\circ\): \[C = 180^\circ - A - B\]

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Law of Cosines

The Law of Cosines relates the lengths of sides of a triangle to the cosine of one of its angles. It is useful for finding unknown sides or angles when two sides and the included angle or three sides are known. The formula is c² = a² + b² - 2ab·cos(C), allowing calculation of missing elements in oblique triangles.

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Intro to Law of Cosines

Law of Sines

The Law of Sines states that the ratios of the lengths of sides to the sines of their opposite angles are equal in any triangle: a/sin(A) = b/sin(B) = c/sin(C). This law helps find unknown angles or sides when given an angle-side pair and another angle or side, especially in non-right triangles.

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Intro to Law of Sines

Triangle Angle Sum Property

The Triangle Angle Sum Property states that the sum of the interior angles of any triangle is always 180°. Knowing one or two angles allows calculation of the remaining angle, which is essential for solving triangles completely when combined with side length information.

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