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 2.5.18

Chapter 3, Problem 2.5.18

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)

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Identify the position of the airplane relative to the radar station at the origin. The coordinates given are (2, 2), which means the airplane is 2 units east and 2 units north of the radar station.

Calculate the angle \( \theta \) that the line from the origin to the point (2, 2) makes with the positive x-axis (east direction) using the tangent function: \( \tan(\theta) = \frac{y}{x} = \frac{2}{2} \).

Find \( \theta \) by taking the arctangent (inverse tangent) of \( \frac{2}{2} \), which gives the angle in degrees or radians measured counterclockwise from the positive x-axis.

Express the bearing in the first method (the compass bearing) by converting the angle \( \theta \) to a compass bearing, which is typically measured clockwise from the north (positive y-axis). Since \( \theta \) is measured from east, use the relationship: \( \text{bearing} = 90^\circ - \theta \).

Express the bearing in the second method (the azimuth bearing) as the angle \( \theta \) measured clockwise from the north direction, or equivalently, the angle from the positive y-axis to the point, which can be found by adjusting \( \theta \) accordingly.

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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 describe direction relative to a reference direction, 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 (e.g., N45°E). Understanding both methods is essential to convert and interpret directions accurately.

Rectangular Coordinate System and Position Vectors

In a rectangular coordinate system, points are represented by (x, y) coordinates relative to the origin. The position vector from the origin to a point defines the direction and distance of the object. This system helps translate spatial locations into angles and distances, which are necessary for calculating bearings.

Calculating Angles Using Trigonometric Functions

To find the bearing from coordinates, use trigonometric functions like tangent to calculate the angle between the position vector and the reference axis (usually the positive y-axis for north). The arctangent function helps determine the angle from the x and y coordinates, which can then be converted into the appropriate bearing format.

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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.

cos θ = 0.85536428

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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

Solve each problem. See Examples 1 and 2. Distance between Two Cities The bearing from Atlanta to Macon is S 27° E, and the bearing from Macon to Augusta is N 63° E. An automobile traveling at 62 mph needs 1¼ hr to go from Atlanta to Macon and 1¾ hr to go from Macon to Augusta. Find the distance from Atlanta to Augusta.

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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. (-4, 0)

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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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