# Right Triangles, Bearings, and other Applications

## Directions based off compass points.

Estimated7 minsto complete
%
Progress
Practice Right Triangles, Bearings, and other Applications

MEMORY METER
This indicates how strong in your memory this concept is
Progress
Estimated7 minsto complete
%
Right Triangles, Bearings, and other Applications
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\begin{align*}30^\circ\end{align*} toward the South of the direction of East. This is represented as E30S\begin{align*}E 30^\circ S\end{align*}. 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?

### 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, N70E\begin{align*}N70^\circ E\end{align*} refers to an angle from the north, towards the east, while N70W\begin{align*}N70^\circ W\end{align*} refers to an angle from the north, towards the west. N70E\begin{align*}N70^\circ E\end{align*} is the same as the angle shown in the graph above. N70W\begin{align*}N70^\circ W\end{align*} would result in an angle in the second quadrant.

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

A ship travels on a N50E\begin{align*}N50^\circ E\end{align*} 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?

The angle between d\begin{align*}d\end{align*} and 10 nm is the complement of 50\begin{align*}50^\circ\end{align*}, which is 40\begin{align*}40^\circ\end{align*}. Therefore we can find d\begin{align*}d\end{align*} using the cosine function:

cos40cos40dcos40d=adjacenthypotenuse=10d=10d=10=10cos4013.05 nm\begin{align*}\cos 40^\circ & = \frac{adjacent}{hypotenuse} = \frac{10}{d}\\ \cos 40^\circ & = \frac{10}{d}\\ d \cos 40^\circ & = 10\\ d & = \frac{10}{\cos 40^\circ} \approx 13.05 \ nm\end{align*}

An airplane flies on a course of S30E\begin{align*}S30^\circ E\end{align*}, for 150 km. How far south is the plane from where it originated?

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\begin{align*}\cos 30^\circ & = \frac{adjacent}{hypotenuse} = \frac{y}{150}\\ \cos 30^\circ & = \frac{y}{150}\\ 150 \cos 30^\circ & = y\\ y & = 150 \cos 30^\circ \approx 130 km\end{align*}

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.

tanθtanθθ=oppositeadjacent=100200=100200=26.57\begin{align*}\tan \theta & = \frac{opposite}{adjacent} = \frac{100}{200}\\ \tan \theta & = \frac{100}{200}\\ \theta & = 26.57^\circ \end{align*}

One way of describing her straight line path is how far north of east she is: E26.57N\begin{align*}E26.57^\circ N\end{align*}

Also, since we know the angle between the East and the North directions is 90\begin{align*}90^\circ\end{align*}, her motion can also be described by how far east of north she is: N63.43E\begin{align*}N63.43^\circ E\end{align*}

### 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:

sin30=opposite5opposite=5sin30opposite=(5)(.5)=2.5\begin{align*} \sin 30^\circ = \frac{opposite}{5}\\ opposite = 5 \sin 30^\circ\\ opposite = (5)(.5) = 2.5\\ \end{align*}

You are 2.5 miles South of where you started.

#### Example 2

Plot a course of S30W\begin{align*}S30^\circ W\end{align*} on a rectangular coordinate system.

#### Example 3

Scott is boating on a course of N15E\begin{align*}N15^\circ E\end{align*}. What course would he need to take to return to where he came from?

The opposite direction would return him to his starting point. This would be S15W\begin{align*}S15^\circ W\end{align*}.

#### Example 4

Adam hikes on a course of N47E\begin{align*}N47^\circ E\end{align*} for 7 km. How far East is Adam from where he started?

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\begin{align*} \sin 47^\circ = \frac{x}{7}\\ x = 7 \sin 47^\circ\\ x = (7)(.7313)\\ x =5.1191 \end{align*}

He is 5.1191 km East of where he started.

### Review

1. Plot a course of N40E\begin{align*}N40^\circ E\end{align*} on a rectangular coordinate system.
2. Plot a course of E30N\begin{align*}E30^\circ N\end{align*} on a rectangular coordinate system.
3. Plot a course of S70W\begin{align*}S70^\circ W\end{align*} on a rectangular coordinate system.
4. Plot a course of W85S\begin{align*}W85^\circ S\end{align*} on a rectangular coordinate system.
5. Plot a course of N42W\begin{align*}N42^\circ W\end{align*} on a rectangular coordinate system.
6. You are on a course of E35N\begin{align*}E35^\circ N\end{align*}. What course would you need to take to return to where you came from?
7. You are on a course of W56S\begin{align*}W56^\circ S\end{align*}. What course would you need to take to return to where you came from?
8. You are on a course of N72W\begin{align*}N72^\circ W\end{align*}. What course would you need to take to return to where you came from?
9. You are on a course of S10E\begin{align*}S10^\circ E\end{align*}. What course would you need to take to return to where you came from?
10. You are on a course of W65N\begin{align*}W65^\circ N\end{align*}. What course would you need to take to return to where you came from?
11. You are on a course of N47E\begin{align*}N47^\circ E\end{align*} for 5 km. How far East are you from where you started?
12. You are on a course of S32E\begin{align*}S32^\circ E\end{align*} for 8 km. How far East are you from where you started?
13. You are on a course of N15W\begin{align*}N15^\circ W\end{align*} for 10 km. How far West are you from where you started?
14. You are on a course of S3W\begin{align*}S3^\circ W\end{align*} for 12 km. How far West are you from where you started?
15. You are on a course of S67E\begin{align*}S67^\circ E\end{align*} for 6 km. How far East are you from where you started?

### Notes/Highlights Having trouble? Report an issue.

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