Ch. 4 - Laws of Sines and Cosines; Vectors

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 4 - Laws of Sines and Cosines; Vectors

Problem 2

Chapter 4, Problem 2

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

Verified step by step guidance

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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Key Concepts

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

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.

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Solving SSA Triangles ("Ambiguous" Case)

Related Practice

Textbook Question

In Exercises 1–4, u and v have the same direction. In each exercise: Is u = v? Explain.

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In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = 5i - 4j, w = -2i - j

Textbook Question

In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.

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Textbook Question

In oblique triangle ABC, C = 68°, a = 5, and b = 6. Find c to the nearest tenth.

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Textbook Question

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

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Textbook Question

In Exercises 1–4, u and v have the same direction. In each exercise: Find ||v||.

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