Textbook Question
Solve each triangle. See Examples 2 and 3.
a = 3.0 ft, b = 5.0 ft, c = 6.0 ft
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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 34
Two forces of 128 lb and 253 lb act on a point. The resultant force is 320 lb. Find the angle between the forces.
Verified step by step guidance1
Identify the given quantities: two forces with magnitudes \(F_1 = 128\) lb and \(F_2 = 253\) lb, and their resultant force \(R = 320\) lb. We need to find the angle \(\theta\) between these two forces.
Recall the law of cosines for vectors, which relates the magnitudes of two vectors and their resultant:
\(R^2 = F_1^2 + F_2^2 + 2 F_1 F_2 \cos(\theta)\)
Substitute the known values into the equation:
\(320^2 = 128^2 + 253^2 + 2 \times 128 \times 253 \times \cos(\theta)\)
Rearrange the equation to isolate \(\cos(\theta)\):
\(\cos(\theta) = \frac{320^2 - 128^2 - 253^2}{2 \times 128 \times 253}\)
Calculate the right-hand side value (without final numeric evaluation here), then use the inverse cosine function to find the angle:
\(\theta = \cos^{-1}(\text{value from previous step})\)
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Resultant of Two Forces
The resultant force is the single force that has the same effect as the two given forces acting together. It can be found using vector addition, considering both magnitude and direction of the forces.
Recommended video:
5:28
Equations with Two Variables
Law of Cosines in Vector Addition
The law of cosines relates the magnitudes of two vectors and the angle between them to the magnitude of their resultant. It is expressed as R² = A² + B² - 2AB cos(θ), where θ is the angle between the forces.
Recommended video:
4:35Intro to Law of Cosines
Solving for the Angle Between Forces
By rearranging the law of cosines formula, the angle between two forces can be found using θ = cos⁻¹((A² + B² - R²) / (2AB)). This allows determination of the angle when the magnitudes and resultant are known.
Recommended video:
04:33Find the Angle Between Vectors
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?
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Textbook Question
Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.
v - u
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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?
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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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