Textbook Question
Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in each figure. Round answers to the nearest tenth.
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Ch. 7 - Applications of Trigonometry and Vectors
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 8, Problem 23
Solve each triangle ABC that exists.
B = 72.2°, b = 78.3 m, c = 145 m
Verified step by step guidance1
Identify the given elements of the triangle: angle \(B = 72.2^\circ\), side \(b = 78.3\) m (opposite angle \(B\)), and side \(c = 145\) m (opposite angle \(C\)). We need to find the remaining parts of the triangle: angles \(A\) and \(C\), and side \(a\).
Use the Law of Sines to find angle \(C\). The Law of Sines states: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). From this, we can write \(\sin C = \frac{c \sin B}{b}\).
Calculate \(\sin C\) using the values of \(b\), \(c\), and \(\sin B\). Then find angle \(C\) by taking the inverse sine (arcsin) of \(\sin C\). Remember to consider the possibility of two solutions for angle \(C\) (the ambiguous case) because sine is positive in two quadrants.
Once angle \(C\) is found, calculate angle \(A\) using the fact that the sum of angles in a triangle is \(180^\circ\): \(A = 180^\circ - B - C\).
Finally, use the Law of Sines again to find side \(a\): \(a = \frac{b \sin A}{\sin B}\). This completes 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.
Law of Sines
The Law of Sines relates the sides and angles of a triangle through the ratio sin(A)/a = sin(B)/b = sin(C)/c. It is essential for solving triangles when given two sides and an angle not included between them, allowing calculation of unknown angles or sides.
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4:27
Intro to Law of Sines
Triangle Angle Sum Property
The sum of the interior angles in any triangle is always 180°. This property helps find the third angle once two angles are known, which is crucial for completing the solution of the triangle.
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4:47Sum and Difference of Tangent
Ambiguous Case of the Law of Sines (SSA Condition)
When two sides and a non-included angle are given (SSA), there can be zero, one, or two possible triangles. Understanding this ambiguity is important to determine all possible solutions or to conclude if no triangle exists.
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9:50Solving SSA Triangles ("Ambiguous" Case)
Related Practice
Textbook Question
Solve each triangle ABC.
A = 39.70°, C = 30.35°, b = 39.74 m
Textbook Question
Write each vector in the form 〈a, b〉. Write answers using exact values or to four decimal places, as appropriate.
Textbook Question
Write each vector in the form 〈a, b〉. Write answers using exact values or to four decimal places, as appropriate.
Textbook Question
For each pair of vectors u and v with angle θ between them, sketch the resultant.
|u| = 20, |v| = 30, θ = 30°
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Textbook Question
Write each vector in the form 〈a, b〉. Write answers using exact values or to four decimal places, as appropriate.