Ch. 2 - Acute Angles and Right Triangles

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

Problem 2.5.11

Chapter 3, Problem 2.5.11

Concept Check The two methods of expressing bearing can be interpreted using a rectangular coordinate system. Suppose that an observer for a radar station is located at the origin of a coordinate system. Find the bearing of an airplane located at each point. Express the bearing using both methods. (-4, 0)

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Identify the position of the airplane relative to the radar station at the origin. The coordinates given are (-4, 0), which means the airplane is 4 units to the left on the x-axis and 0 units on the y-axis.

Recall that bearings are typically measured clockwise from the north direction (positive y-axis). The first method expresses bearing as an angle clockwise from north, ranging from 0° to 360°.

Determine the angle of the point (-4, 0) relative to the positive y-axis. Since the point lies on the negative x-axis, the angle from the positive y-axis clockwise to the point is 270°.

The second method expresses bearing using the format 'N or S [angle] E or W', which means starting from north or south, measure an angle east or west. For the point (-4, 0), since it lies directly west of the origin, the bearing is 'N 90° W'.

Summarize the two bearings: the first method gives a bearing of 270°, and the second method gives a bearing of 'N 90° W'.

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Key Concepts

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

Bearing and Its Methods

Bearing is a way to express direction relative to a reference point, usually north. The two common methods are the compass bearing, measured clockwise from north (0° to 360°), and the quadrant bearing, expressed as an angle east or west of north or south. Understanding both methods allows conversion between them and clear communication of directions.

Rectangular Coordinate System and Direction

In a rectangular coordinate system, points are located by (x, y) coordinates relative to the origin. The x-axis represents east-west direction, and the y-axis represents north-south. Interpreting bearings from coordinates involves relating these axes to compass directions, where positive y is north and positive x is east.

Calculating Bearing from Coordinates

To find the bearing of a point from the origin, calculate the angle formed by the line connecting the origin to the point and the north direction. This involves using inverse trigonometric functions like arctangent to find the angle relative to the axes, then converting it to the appropriate bearing format.

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Related Practice

Textbook Question

Find a value of θ in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. See Example 2.

cos θ = 0.85536428

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Textbook Question

Concept Check The two methods of expressing bearing can be interpreted using a rectangular coordinate system. Suppose that an observer for a radar station is located at the origin of a coordinate system. Find the bearing of an airplane located at each point. Express the bearing using both methods. (2, 2)

Textbook Question

(Modeling) Fish's View of the World The figure shows a fish's view of the world above the surface of the water. (Data from Walker, J., 'The Amateur Scientist,' Scientific American.) Suppose that a light ray comes from the horizon, enters the water, and strikes the fish's eye. Assume that this ray gives a value of 90° for angle θ₁ in the formula for Snell's law. (In a practical situation, this angle would probably be a little less than 90°.) The speed of light in water is about 2.254 x 10⁸ m per sec. Find angle θ₂ to the nearest tenth.

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Textbook Question

Find a value of θ in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. See Example 2.

csc θ = 1.3861147

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Textbook Question

Find a value of θ in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. See Example 2.

sin θ = 0.84802194

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Textbook Question

Find two angles in the interval [0°, 360°) that satisfy each of the following. Round answers to the nearest degree. tan θ = 0.70020753

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