Textbook Question
Write each vector in the form 〈a, b〉. Write answers using exact values or to four decimal places, as appropriate.
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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 25
Solve each triangle. See Examples 2 and 3.
a = 42.9 m, b = 37.6 m, c = 62.7 m
Verified step by step guidance1
Identify the given sides of the triangle: \(a = 42.9\) m, \(b = 37.6\) m, and \(c = 62.7\) m. Since all three sides are known, this is a side-side-side (SSS) triangle problem.
Use the Law of Cosines to find one of the angles. For example, to find angle \(C\) opposite side \(c\), apply the formula: \[\cos C = \frac{a^2 + b^2 - c^2}{2ab}\]
Calculate \(\cos C\) using the given side lengths, then find angle \(C\) by taking the inverse cosine (arccos) of that value: \[C = \arccos\left(\frac{a^2 + b^2 - c^2}{2ab}\right)\]
Next, use the Law of Cosines again to find another angle, such as angle \(A\) opposite side \(a\): \[\cos A = \frac{b^2 + c^2 - a^2}{2bc}\] and then \[A = \arccos\left(\frac{b^2 + c^2 - a^2}{2bc}\right)\]
Finally, find the third angle \(B\) by using the fact that the sum of angles in a triangle is \(180^\circ\): \[B = 180^\circ - A - C\]
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Triangle Side Lengths and Classification
Understanding the given side lengths helps classify the triangle (scalene, isosceles, or equilateral) and determines which methods to use for solving it. Here, all sides are different, indicating a scalene triangle, which requires specific laws to find angles.
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4:18
Finding Missing Side Lengths
Law of Cosines
The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is essential for finding unknown angles when all three sides are known, using the formula c² = a² + b² - 2ab cos(C).
Recommended video:
4:35Intro to Law of Cosines
Triangle Angle Sum Property
The sum of the interior angles in any triangle is always 180 degrees. After finding one or two angles using the Law of Cosines, this property allows calculation of the remaining angle to fully solve the triangle.
Recommended video:
4:47Sum and Difference of Tangent
Related Practice
Textbook Question
Solve each triangle. See Examples 2 and 3.
a = 9.3 cm, b = 5.7 cm, c = 8.2 cm
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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Textbook Question
Write each vector in the form 〈a, b〉. Write answers using exact values or to four decimal places, as appropriate.
Textbook Question
Solve each triangle ABC that exists.
A = 38° 40', a = 9.72 m, b = 11.8 m
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Textbook Question
Solve each triangle ABC.
B = 42.88°, C = 102.40°, b = 3974 ft
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