Textbook Question
Use the figure to find each vector: - u. Use vector notation as in Example 4.
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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 33
Two forces of 692 newtons and 423 newtons act on a point. The resultant force is 786 newtons. Find the angle between the forces.
Verified step by step guidance1
Identify the given values: the magnitudes of the two forces are \(F_1 = 692\) newtons and \(F_2 = 423\) newtons, and the magnitude of the resultant force is \(R = 786\) newtons.
Recall the formula for the magnitude of the resultant of two forces acting at an angle \(\theta\) between them:
\(R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos \theta}\)
Substitute the known values into the formula:
\(786 = \sqrt{692^2 + 423^2 + 2 \times 692 \times 423 \times \cos \theta}\)
Square both sides to eliminate the square root:
\(786^2 = 692^2 + 423^2 + 2 \times 692 \times 423 \times \cos \theta\)
Rearrange the equation to solve for \(\cos \theta\):
\(\cos \theta = \frac{786^2 - 692^2 - 423^2}{2 \times 692 \times 423}\)
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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 two or more forces acting together. When two forces act at a point, their resultant can be found using vector addition, which depends on the magnitudes of the forces and the angle between them.
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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 states that R² = A² + B² + 2AB cos(θ), where R is the resultant, A and B are the forces, and θ is the angle between them.
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4:35Intro to Law of Cosines
Solving for the Angle Between Forces
To find the angle between two forces given their magnitudes and resultant, rearrange the law of cosines formula to solve for cos(θ). Then use the inverse cosine function to determine the angle, ensuring the correct interpretation of the angle in the context of the problem.
Recommended video:
04:33Find the Angle Between Vectors
Related Practice
Textbook Question
Use the figure to find each vector: u - v. Use vector notation as in Example 4.
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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
Solve each triangle. See Examples 2 and 3.
A = 112.8°, b = 6.28 m, c = 12.2 m
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Textbook Question
Two rescue vessels are pulling a broken-down motorboat toward a boathouse with forces of 840 lb and 960 lb. The angle between these forces is 24.5°. Find the direction and magnitude of the equilibrant.
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Textbook Question
Without using the law of sines, explain why no triangle ABC can exist that satisfies A = 103° 20', a = 14.6 ft, b = 20.4 ft.
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