Textbook Question
Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.
-5v
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 35
Solve each triangle. See Examples 2 and 3.
a = 3.0 ft, b = 5.0 ft, c = 6.0 ft
Verified step by step guidance1
Identify the given sides of the triangle: \(a = 3.0\) ft, \(b = 5.0\) ft, and \(c = 6.0\) ft. 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 \(c\), use the formula: \[\cos C = \frac{a^2 + b^2 - c^2}{2ab}\]
Calculate \(\cos C\) by substituting the known side lengths into the formula, then find angle \(C\) by taking the inverse cosine (arccos) of that value: \[C = \cos^{-1}\left(\frac{a^2 + b^2 - c^2}{2ab}\right)\]
Repeat the Law of Cosines to find another angle, for example angle \(B\) opposite side \(b\(, using: \[\cos B = \frac{a^2 + c^2 - b^2}{2ac}\] and then find \)B = \cos^{-1}(\text{value})\).
Find the third angle \(A\) using the fact that the sum of angles in a triangle is \(180^\circ\): \[A = 180^\circ - B - C\].
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 especially useful for solving triangles when all three sides are known, allowing calculation of each angle using the formula: c² = a² + b² - 2ab·cos(C).
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Intro to Law of Cosines
Triangle Classification and Properties
Understanding the types of triangles (acute, obtuse, or right) based on side lengths helps in interpreting results. The triangle inequality theorem ensures the given sides can form a triangle, and recognizing side relationships aids in predicting angle sizes.
Recommended video:
4:42Review of Triangles
Angle Sum Property of Triangles
The sum of the interior angles in any triangle is always 180 degrees. After finding two angles using the Law of Cosines, the third angle can be found by subtracting their sum from 180°, completing the solution of the triangle.
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4:47Sum and Difference of Tangent
Related Practice
Textbook Question
A ship is sailing due north. At a certain point the bearing of a lighthouse 12.5 km away is N 38.8° E. Later on, the captain notices that the bearing of the lighthouse has become S 44.2° E. How far did the ship travel between the two observations of the lighthouse?
Textbook Question
To determine the distance RS across a deep canyon, Rhonda lays off a distance TR = 582 yd. She then finds that T = 32° 50' and R = 102° 20'. Find RS. See the figure.
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Textbook Question
Apply the law of sines to the following:
A = 104°, a = 26.8, b = 31.3.
What happens when we try to find the measure of angle B using a calculator?
1
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Textbook Question
Radio direction finders are placed at points A and B, which are 3.46 mi apart on an east-west line, with A west of B. From A the bearing of a certain radio transmitter is 47.7°, and from B the bearing is 302.5°. Find the distance of the transmitter from A.
Textbook Question
A force of 176 lb makes an angle of 78° 50′ with a second force. The resultant of the two forces makes an angle of 41° 10′ with the first force. Find the magnitudes of the second force and of the resultant.
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