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. (0, -2)

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Understand that the observer is at the origin (0,0) and the airplane is at the point (0, -2) in the coordinate plane. The bearing is the direction from the observer to the airplane.

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 vector from the origin to the point (0, -2). Since the point lies directly on the negative y-axis, the direction is straight south.

Express the bearing in the first method: since the airplane is directly south, the bearing is 180° clockwise from north.

For the second method, bearings are expressed as angles east or west of north or south. Since the airplane is directly south, the bearing is simply S 0° E (or just due south).

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주요 개념

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Bearing and Its Methods

Bearing is a way to describe direction relative to a reference, 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 for accurate navigation.

Rectangular Coordinate System and Position Vectors

In a rectangular coordinate system, points are located using (x, y) coordinates relative to the origin. The position vector from the origin to a point can be used to determine direction and distance. This system helps translate spatial locations into angles and bearings by relating coordinates to directions.

Calculating Angles Using Inverse Trigonometric Functions

To find the bearing from coordinates, inverse trigonometric functions like arctangent are used to calculate the angle between the position vector and a reference axis. Adjustments are made based on the quadrant to get the correct bearing angle, ensuring the direction is accurately represented relative to north.

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교과서 질문

(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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교과서 질문

Solve each problem. See Examples 1 and 2. Distance Traveled by a Ship A ship travels 55 km on a bearing of 27° and then travels on a bearing of 117° for 140 km. Find the distance from the starting point to the ending point.

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교과서 질문

CONCEPT PREVIEW Match each trigonometric function value or angle in Column I with its appropriate approximation in Column II.

Column I: 1.

cot⁻¹ 30

Column II:

A. 88.09084757°

B. 63.25631605°

C. 1.909152433°

D. 17.45760312°

E. 0.2867453858