### Bearings

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,

Now, let's look at a problem where we find the distance using right triangles as bearings.

A ship travels on a

The angle between

An airplane flies on a course of

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

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 section?

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

One way of describing her straight line path is how far north of east she is:

Also, since we know the angle between the East and the North directions is

### Examples

#### Example 1

Earlier, you were asked is it possible to determine how far South you are from where you started?

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:

You are 2.5 miles South of where you started.

#### Example 2

Plot a course of

#### Example 3

Scott is boating on a course of

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

#### Example 4

Adam hikes on a course of

** **

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

He is 5.1191 km East of where he started.

### Review

- Plot a course of
N40∘E on a rectangular coordinate system. - Plot a course of
E30∘N on a rectangular coordinate system. - Plot a course of
S70∘W on a rectangular coordinate system. - Plot a course of
W85∘S on a rectangular coordinate system. - Plot a course of
N42∘W on a rectangular coordinate system. - You are on a course of
E35∘N . What course would you need to take to return to where you came from? - You are on a course of
W56∘S . What course would you need to take to return to where you came from? - You are on a course of
N72∘W . What course would you need to take to return to where you came from? - You are on a course of
S10∘E . What course would you need to take to return to where you came from? - You are on a course of
W65∘N . What course would you need to take to return to where you came from? - You are on a course of
N47∘E for 5 km. How far East are you from where you started? - You are on a course of
S32∘E for 8 km. How far East are you from where you started? - You are on a course of
N15∘W for 10 km. How far West are you from where you started? - You are on a course of
S3∘W for 12 km. How far West are you from where you started? - You are on a course of
S67∘E for 6 km. How far East are you from where you started?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.14.