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.
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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 36
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.
Verified step by step guidance1
Draw a diagram to visualize the problem: place points A and B on a horizontal east-west line with A to the west and B to the east, 3.46 miles apart. Mark the bearings from A and B to the transmitter as given.
Convert the bearings into angles relative to the east-west line. From A, the bearing 47.7° is measured clockwise from north, so find the angle between the line AB and the line from A to the transmitter. Similarly, convert the bearing from B (302.5°) into an angle relative to the line AB.
Use the angles found to determine the directions of the lines from A and B to the transmitter. This will allow you to form a triangle with vertices at A, B, and the transmitter.
Apply the Law of Sines to the triangle formed by points A, B, and the transmitter. You know the side AB = 3.46 miles and the two angles at A and B (found from the bearings). Use the Law of Sines formula: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\), where sides are opposite their respective angles.
Solve for the side opposite the angle at B, which corresponds to the distance from A to the transmitter. This will give you the required distance without calculating the final numeric value.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Bearings and Angle Measurement
Bearings are angles measured clockwise from the north direction to the line connecting the observer to the object. Understanding how to interpret and convert bearings into standard angles or directions is essential for locating positions on a plane, especially in navigation and surveying problems.
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Reference Angles on the Unit Circle
Triangle Formation and Law of Sines
The positions of points A, B, and the transmitter form a triangle. The Law of Sines relates the sides and angles of a triangle, allowing calculation of unknown distances or angles when some are known. This law is crucial for solving the distance from A to the transmitter using the given bearings.
Recommended video:
4:27Intro to Law of Sines
Coordinate Geometry and Vector Representation
Representing points and directions using coordinate geometry helps visualize the problem. By placing A and B on an east-west axis and using bearings to determine directions, one can translate the problem into a coordinate system, facilitating the use of trigonometric relationships to find distances.
Recommended video:
05:29Adding Vectors Geometrically
Related Practice
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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Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.
-5v
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A force of 25 lb is required to hold an 80-lb crate on a hill. What angle does the hill make with the horizontal?
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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
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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