In Exercises 9–24, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. b = 5, c = 3, A = 102°

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Identify the given values: side \( b = 5 \), side \( c = 3 \), and angle \( A = 102^\circ \).

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

Substitute the known values into the Law of Cosines formula: \( a^2 = 5^2 + 3^2 - 2 \cdot 5 \cdot 3 \cdot \cos(102^\circ) \).

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

Use the Law of Sines to find another angle, say \( B \): \( \frac{\sin(B)}{b} = \frac{\sin(A)}{a} \).

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Law of Sines

The Law of Sines is a fundamental principle in trigonometry that relates the ratios of the lengths of sides of a triangle to the sines of its angles. It states that for any triangle, the ratio of a side length to the sine of its opposite angle is constant. This law is particularly useful for solving triangles when given two angles and a side (AAS or ASA) or two sides and a non-included angle (SSA).

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Intro to Law of Sines

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 some angles are already known. In the context of the given problem, knowing one angle allows us to calculate the remaining angles, which is crucial for solving the triangle.

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Solving Right Triangles with the Pythagorean Theorem

Ambiguous Case of the Law of Sines

The Ambiguous Case of the Law of Sines occurs when using the SSA condition (two sides and a non-included angle) to solve a triangle. This situation can lead to zero, one, or two possible triangles, depending on the given measurements. Understanding this concept is vital for correctly interpreting the results when solving triangles, especially when the given angle is obtuse, as in this problem.

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