Math Problem Statement

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 the point ​(negative 10​,0​). Express the bearing using both methods.

Solution

In a rectangular coordinate system, bearings can be expressed in two ways:

  1. Clockwise from North (standard bearing)
  2. Counterclockwise from the positive x-axis (also called azimuth or polar coordinates)

Given that the airplane is located at the point (10,0)(-10, 0), this means the airplane is 10 units directly to the left of the origin, lying on the negative x-axis.

Method 1: Bearing Clockwise from North (Standard Bearing)

In this method, we measure the angle clockwise from the north (positive y-axis). Since the point (10,0)(-10, 0) is directly west of the origin, the bearing is 270° from North.

Method 2: Bearing Counterclockwise from the Positive X-axis (Azimuth)

In this method, we measure the angle counterclockwise from the positive x-axis. Since the airplane is located directly on the negative x-axis, the azimuth is 180°.

Final Answer:

  • Standard bearing (clockwise from North): 270°
  • Azimuth (counterclockwise from positive x-axis): 180°

Would you like further details on how these angles are measured or any clarifications?

Here are five related questions you might consider:

  1. What is the bearing of an object located at (10,0)(10, 0)?
  2. How would you calculate the bearing of a point at (3,4)(3, 4)?
  3. What are some practical applications of using standard bearings in navigation?
  4. How does the concept of a unit circle relate to these bearings?
  5. Can bearings ever be negative, and if so, how would they be interpreted?

Tip: Bearings always need to be described within the proper reference framework, so pay attention to whether you're working with standard or azimuth bearings.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Bearings
Polar Coordinates

Formulas

Bearing (clockwise from North)
Azimuth (counterclockwise from positive x-axis)

Theorems

Angle measurement in polar coordinates
Bearing calculation in navigation

Suitable Grade Level

Grades 9-11