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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, 9780135440780, 9780136881117, 9780135440827, 9780135924891
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TOC.tableOfContents
- 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
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Capitolo 8, Problema 35
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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Guida verificata passo dopo passo1
Convert all given angles from degrees and minutes to decimal degrees for easier calculation. For example, 78° 50′ becomes 78 + 50/60 degrees, and similarly for 41° 10′.
Label the forces: let the first force be \( F_1 = 176 \) lb, the second force be \( F_2 \), and the resultant force be \( R \). The angle between \( F_1 \) and \( F_2 \) is \( \theta = 78.8333^\circ \) (converted from 78° 50′), and the angle between \( F_1 \) and \( R \) is \( \alpha = 41.1667^\circ \) (converted from 41° 10′).
Use the Law of Cosines to express the magnitude of the resultant force \( R \) in terms of \( F_1 \), \( F_2 \), and \( \theta \):
\[
R^2 = F_1^2 + F_2^2 + 2 F_1 F_2 \cos(\theta)
\]
Use the Law of Sines or the formula for the angle between two vectors to relate the given angle \( \alpha \) to the forces. Specifically, use the formula for the tangent of the angle between \( F_1 \) and \( R \):
\[
\tan(\alpha) = \frac{F_2 \sin(\theta)}{F_1 + F_2 \cos(\theta)}
\]
This equation allows you to solve for \( F_2 \).
Solve the equation from step 4 for \( F_2 \), then substitute \( F_2 \) back into the Law of Cosines equation from step 3 to find the magnitude of the resultant force \( R \).
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Concetti chiave
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Vector Addition of Forces
When two forces act at an angle, their combined effect is found by vector addition. The resultant force is the vector sum of the individual forces, and its magnitude and direction depend on both the magnitudes and the angle between the forces.
Video consigliato:
Percorso guidato
05:29
Adding Vectors Geometrically
Law of Cosines
The Law of Cosines relates the lengths of sides of a triangle to the cosine of one of its angles. It is used to find the magnitude of the resultant force when two forces and the angle between them are known, or to find an unknown force when the resultant and angles are given.
Video consigliato:
Percorso guidato
4:35Intro to Law of Cosines
Law of Sines
The Law of Sines relates the ratios of the lengths of sides of a triangle to the sines of their opposite angles. It helps determine unknown angles or side lengths in the force triangle, especially when the resultant's angle relative to one force is given.
Video consigliato:
Percorso guidato
4:27Intro to Law of Sines
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Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.
-5v
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Domanda del libro di testo
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Domanda del libro di testo
A force of 28.7 lb makes an angle of 42° 10′ with a second force. The resultant of the two forces makes an angle of 32° 40′ with the first force. Find the magnitudes of the second force and of the resultant.
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Domanda del libro di testo
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.
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