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 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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Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
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.
Recommended video:
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.
Recommended video:
4:27Intro to Law of Sines
Related Practice
Textbook Question
Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.
-5v
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
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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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
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.