In Exercises 9–24, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. a = 5, b = 7, C = 42°

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Identify the given values: side \( a = 5 \), side \( b = 7 \), and angle \( C = 42^\circ \).

Use the Law of Cosines to find side \( c \): \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \).

Substitute the known values into the Law of Cosines formula: \( c^2 = 5^2 + 7^2 - 2 \cdot 5 \cdot 7 \cdot \cos(42^\circ) \).

Calculate \( c \) by taking the square root of the result from the previous step.

Use the Law of Sines to find one of the remaining angles, say \( A \): \( \frac{\sin(A)}{a} = \frac{\sin(C)}{c} \).

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Key Concepts

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

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 is expressed as a/sin(A) = b/sin(B) = c/sin(C). It is particularly useful for solving triangles when two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA) are known.

Triangle Sum Theorem

The Triangle Sum Theorem states that the sum of the interior angles of a triangle is always 180 degrees. This theorem is essential for finding unknown angles in a triangle when two angles are known, allowing for the calculation of the third angle by subtracting the sum of the known angles from 180 degrees.

Side-Angle Relationships

In any triangle, the lengths of the sides are directly related to the measures of the angles opposite those sides. Specifically, the larger the angle, the longer the side opposite it. This relationship is crucial when using the Law of Sines or the Law of Cosines to solve for unknown sides or angles in a triangle.

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