Textbook Question
The bearing of a lighthouse from a ship was found to be N 37° E. After the ship sailed 2.5 mi due south, the new bearing was N 25° E. Find the distance between the ship and the lighthouse at each location.
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 36
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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Verified step by step guidance1
Convert all given angles from degrees and minutes to decimal degrees for easier calculation. For example, 42° 10′ becomes \(42 + \frac{10}{60}\) degrees, and similarly for 32° 40′.
Let the magnitude of the second force be \(F_2\) and the magnitude of the resultant force be \(R\). The first force \(F_1\) is given as 28.7 lb.
Use the Law of Cosines to relate the magnitudes of the forces and the angle between them. The angle between the two forces is \$42° 10′$, so write the equation:
\(R^2 = F_1^2 + F_2^2 + 2 F_1 F_2 \cos(42° 10′)\)
Use the Law of Sines to relate the angles and sides in the triangle formed by the two forces and their resultant. The resultant makes an angle of \$32° 40′$ with the first force, so write:
\(\frac{F_2}{\sin(32° 40′)} = \frac{R}{\sin(42° 10′ + 32° 40′)}\)
Solve the system of two equations from steps 3 and 4 simultaneously to find the unknown magnitudes \(F_2\) and \(R\).
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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, considering both magnitude and direction. Understanding how to add vectors graphically or analytically is essential to solve for unknown forces.
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Law of Cosines in Vector Problems
The Law of Cosines relates the magnitudes of two vectors and the angle between them to the magnitude of their resultant. It is used to find unknown sides or angles in non-right triangles formed by force vectors, enabling calculation of the second force or resultant magnitude.
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4:35Intro to Law of Cosines
Angle Relationships Between Forces and Resultant
The angles between forces and their resultant are crucial for setting up equations. Knowing the angle the resultant makes with one force helps determine the direction and magnitude of the other force using trigonometric relationships and vector components.
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04:33Find the Angle Between Vectors
Related Practice
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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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Find the force required to keep a 3000-lb car parked on a hill that makes an angle of 15° with the horizontal.
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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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