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 = 107°, C = 30°, c = 126

검증된 단계별 안내

Identify the given elements of the triangle: angle \(B = 107^\circ\), angle \(C = 30^\circ\), and side \(c = 126\) (opposite angle \(C\)).

Calculate the measure of the third angle \(A\) using the triangle angle sum property: \(A = 180^\circ - B - C = 180^\circ - 107^\circ - 30^\circ\).

Use the Law of Sines to find side \(a\) opposite angle \(A\): \(\frac{a}{\sin A} = \frac{c}{\sin C}\), so \(a = \frac{c \cdot \sin A}{\sin C}\).

Similarly, use the Law of Sines to find side \(b\) opposite angle \(B\): \(\frac{b}{\sin B} = \frac{c}{\sin C}\), so \(b = \frac{c \cdot \sin B}{\sin C}\).

Round the calculated side lengths \(a\) and \(b\) to the nearest tenth and the angle \(A\) to the nearest degree to complete the solution of the triangle.

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주요 개념

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Triangle Angle Sum Theorem

This theorem states that the sum of the interior angles of any triangle is always 180°. Given two angles, you can find the third by subtracting their sum from 180°. This is essential for determining the missing angle when two angles are provided.

Law of Sines

The Law of Sines relates the ratios of sides to the sines of their opposite angles in any triangle: (a/sin A) = (b/sin B) = (c/sin C). It is crucial for solving triangles when given two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).

Ambiguous Case of the Law of Sines (SSA)

When given two sides and a non-included angle (SSA), there may be zero, one, or two possible triangles. This ambiguity arises because the given side can form different triangles depending on its length relative to the height. Recognizing this case helps determine if multiple solutions exist.