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:
- Clockwise from North (standard bearing)
- Counterclockwise from the positive x-axis (also called azimuth or polar coordinates)
Given that the airplane is located at the point , 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 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:
- What is the bearing of an object located at ?
- How would you calculate the bearing of a point at ?
- What are some practical applications of using standard bearings in navigation?
- How does the concept of a unit circle relate to these bearings?
- 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
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