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Ch. 7 - Applications of Trigonometry and Vectors
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 7 - Applications of Trigonometry and VectorsProblem 45
Chapter 8, Problem 45

A ship leaves port on a bearing of 34.0° and travels 10.4 mi. The ship then turns due east and travels 4.6 mi. How far is the ship from port, and what is its bearing from port?

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Draw a diagram to visualize the ship's path: start at the port (origin), draw a line representing the ship's first leg of 10.4 miles at a bearing of 34.0° from north, then from that point draw a second leg 4.6 miles due east.
Resolve the first leg of the journey into its north-south and east-west components using trigonometry: calculate the eastward component as \(10.4 \times \sin(34.0^\circ)\) and the northward component as \(10.4 \times \cos(34.0^\circ)\).
Add the eastward component of the second leg (4.6 miles due east) to the eastward component from the first leg to get the total eastward displacement from the port. The northward displacement remains the same as from the first leg since the second leg is due east.
Use the Pythagorean theorem to find the straight-line distance from the port to the ship's final position: calculate \(\sqrt{(\text{total eastward displacement})^2 + (\text{northward displacement})^2}\).
Find the bearing from the port to the ship's final position by calculating the angle east of north using the inverse tangent function: \(\theta = \tan^{-1}\left(\frac{\text{total eastward displacement}}{\text{northward displacement}}\right)\). This angle is the bearing measured clockwise from north.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Bearing and Direction in Navigation

Bearing is the direction or path along which something moves, measured in degrees clockwise from the north. Understanding how to interpret bearings, such as 34°, is essential for determining the ship's initial direction relative to the port.
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Vector Addition and Displacement

The ship's journey involves two legs with different directions, so its overall displacement is found by vector addition. Breaking each leg into components (using trigonometric functions) allows calculation of the resultant position relative to the starting point.
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Trigonometric Functions and Distance Calculation

Using sine and cosine functions helps resolve distances into perpendicular components. The Pythagorean theorem then calculates the straight-line distance from the port, and inverse trigonometric functions determine the bearing from the port.
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Related Practice
Textbook Question

Starting at point A, a ship sails 18.5 km on a bearing of 189°, then turns and sails 47.8 km on a bearing of 317°. Find the distance of the ship from point A.

Textbook Question

Given vectors u and v, find: v - 3u. 

u = 〈-1, 2〉, v = 〈3, 0〉

Textbook Question

A luxury liner leaves port on a bearing of 110.0° and travels 8.8 mi. It then turns due west and travels 2.4 mi. How far is the liner from port, and what is its bearing from port?

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Textbook Question

Given vectors u and v, find: 2u + 3v. 

u = 〈-1, 2〉, v = 〈3, 0〉

Textbook Question

Write each vector in the form a i + b j.

〈6, -3〉

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Textbook Question

A crate is supported by two ropes. One rope makes an angle of 46° 20′ with the horizontal and has a tension of 89.6 lb on it. The other rope is horizontal. Find the weight of the crate and the tension in the horizontal rope.