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1.14: Right Triangles, Bearings, and other Applications

Difficulty Level: At Grade Created by: CK-12
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While on a camping trip with your friends, you take an orienteering trip. You end up on a course which results in you hiking 30 toward the South of the direction of East. This is represented as E30S. You hike until you are 5 miles from where you started. Is it possible to determine how far South you are from where you started?

Read on, and when you have completed this Concept, you'll be able to make this calculation.

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GCSE Math Bearings

Guidance

We can also use right triangles to find distances using angles given as bearings. In navigation, a bearing is the direction from one object to another. In air navigation, bearings are given as angles rotated clockwise from the north. The graph below shows an angle of 70 degrees:

It is important to keep in mind that angles in navigation problems are measured this way, and not the same way angles are measured in trigonometry. Further, angles in navigation and surveying may also be given in terms of north, east, south, and west. For example, N70E refers to an angle from the north, towards the east, while N70W refers to an angle from the north, towards the west. N70E is the same as the angle shown in the graph above. N70W would result in an angle in the second quadrant.

Example A

A ship travels on a N50E course. The ship travels until it is due north of a port which is 10 nautical miles due east of the port from which the ship originated. How far did the ship travel?

Solution: The angle between d and 10 nm is the complement of 50, which is 40. Therefore we can find d using the cosine function:

cos40cos40dcos40d=adjacenthypotenuse=10d=10d=10=10cos4013.05 nm

Example B

An airplane flies on a course of S30E, for 150 km. How far south is the plane from where it originated?

Solution: We can construct a triangle using the known information, and then use the cosine function to solve the problem:

cos30cos30150cos30y=adjacenthypotenuse=y150=y150=y=150cos30130km

Example C

Jean travels to school each day by walking 200 meters due East, and then turning left and walking 100 meters due North. If she had walked in a straight line, what would the angle between her home and the school be if the beginning of the angle is taken from due East? What would be two different ways to describe the direction to take walking there in a straight line, using what we've learned in this Concept?

Solution: From the triangle given above, we can use the tangent function to determine the angle if she had walked in a straight line.

tanθtanθθ=oppositeadjacent=100200=100200=26.57

One way of describing her straight line path is how far north of east she is: E26.57N

Also, since we know the angle between the East and the North directions is 90, her motion can also be described by how far east of north she is: N63.43E

Guided Practice

1. Plot a course of S30W on a rectangular coordinate system.

2. Scott is boating on a course of N15E. What course would he need to take to return to where he came from?

3. Adam hikes on a course of N47E for 7 km. How far East is Adam from where he started?

Solutions:

1.

2. The opposite direction would return him to his starting point. This would be S15W.

3.

We can find the length of the triangle above (which is how far he traveled East) by using the sine function:

sin47=x7x=7sin47x=(7)(.7313)x=5.1191

He is 5.1191 km East of where he started.

Concept Problem Solution

From your knowledge of how to construct a triangle using bearings, you can draw the following:

This shows that the opposite side of the triangle is what's not known. Therefore, you can use the sine function to solve the problem:

sin30=opposite5opposite=5sin30opposite=(5)(.5)=2.5

You are 2.5 miles South of where you started.

Explore More

  1. Plot a course of N40E on a rectangular coordinate system.
  2. Plot a course of E30N on a rectangular coordinate system.
  3. Plot a course of S70W on a rectangular coordinate system.
  4. Plot a course of W85S on a rectangular coordinate system.
  5. Plot a course of N42W on a rectangular coordinate system.
  6. You are on a course of E35N. What course would you need to take to return to where you came from?
  7. You are on a course of W56S. What course would you need to take to return to where you came from?
  8. You are on a course of N72W. What course would you need to take to return to where you came from?
  9. You are on a course of S10E. What course would you need to take to return to where you came from?
  10. You are on a course of W65N. What course would you need to take to return to where you came from?
  11. You are on a course of N47E for 5 km. How far East are you from where you started?
  12. You are on a course of S32E for 8 km. How far East are you from where you started?
  13. You are on a course of N15W for 10 km. How far West are you from where you started?
  14. You are on a course of S3W for 12 km. How far West are you from where you started?
  15. You are on a course of S67E for 6 km. How far East are you from where you started?

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 1.14. 

Vocabulary

Bearing

Bearing

Bearing is how direction is measured at sea. North is 0^\circ, east is 90^\circ, south is 180^\circ , and west is 270^\circ.

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Difficulty Level:

At Grade

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Date Created:

Sep 26, 2012

Last Modified:

Feb 26, 2015
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