In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle. B = 66°, a = 17, c = 12

Key Concepts

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 one side (AAS or ASA) or two sides and a non-included angle (SSA).

Triangle Existence Criteria

To determine if a triangle can exist with given side lengths and angles, one must consider the triangle inequality theorem and the conditions for angle-side relationships. Specifically, the sum of the angles in a triangle must equal 180 degrees, and the lengths of any two sides must be greater than the length of the third side. If these conditions are not met, it may be concluded that 'no triangle' exists.

Ambiguous Case of SSA

The ambiguous case occurs when two sides and a non-included angle (SSA) are known, leading to the possibility of zero, one, or two triangles. This situation arises because the given angle may correspond to two different configurations of the triangle. To resolve this, one must analyze the given information and apply the Law of Sines to check for multiple solutions or the absence of a triangle.