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Solving Real World Problems With Compass Bearings

Instructor: Matthew Bergstresser

Matthew has a Master of Arts degree in Physics Education

There are several methods to indicate which way an object is heading by using a compass. In this lesson, we will focus on three figure compass bearings to solve real world problems.

Out on the Ocean

Imagine you were taken out on the ocean far from land and put in a boat and told to steer the boat towards land. This might not sound like too much fun unless you have the spirit for adventure! We'll assume it isn't nighttime and you aren't well-versed in navigating by the constellations. One way you could know where to steer the boat is if you knew which direction land was and you had a compass. Let's practice by going through the basics of using a compass and trigonometry and then working some scenarios using bearing angles and trigonometry.

Compass

A compass is a device with a magnetic needle that always points north. The compass is subdivided into 1 or 2° increments starting at due north (000°) and rotating clockwise.


A compass
compass


Since the red portion of the needle always points north, we can gather the imaginary person holding this compass is actually facing 342°, which is 18° west of north. If they rotated the compass clockwise so that 000° is lined up with the red end of the needle, 342° would be aimed in the direction they are facing, which is called a heading.

Right Triangle Trigonometry Review

Now let's review some right triangle trigonometry because we will need these skills to solve real world bearing problems.

There are three trigonometric functions that are used when dealing with right triangles: sine, cosine and tangent. These functions are the ratios of two of the three sides of a right triangle. Diagram 1 shows a right triangle with each side labeled relative to the angle θ.


Diagram 1. A right triangle
triangle


Let's look at the equations for the trigonometric functions:


The trigonometric functions.
trig_functions


Now we have all of the tools we need to solve problems involving compass bearings. Let's get to it!

Real World Compass Bearing Problems

Example 1

Prompt: A sailboat left the dock heading 030° and traveled 100 kilometers. How far east of the dock is the sailboat at this time?

Solution: It is always a good idea to draw a diagram of the scenario. Diagram 2 shows the diagram for this scenario.


Diagram 2. Initial setup of Example 1.
ex1


To determine how far east the sailboat is, we need to determine the side of the triangle directly across from the 30° angle as shown in Diagram 3.


Diagram 3. The green arrow represents the eastward distance of the sailboat.
ex1a


Since we have two pieces of information regarding this right triangle, we can solve for the eastward distance the sailboat traveled. We will use the sine trigonometric function because it is opposite the 30° angle, which looks like:

setup1

Solving for the opposite side we get:

ex1_answer

Therefore the sailboat is 50 km east from where it started.

Example 2

Prompt: A rancher has to mend fences on his ranch. He heads out from the ranch building heading 090° for 1 km. Then he turns and heads 200° for 0.5 km. What heading would someone take to reach the rancher directly?

Solution: Let's start with a sketch of the situation, which is shown in Diagram 4:


Diagram 4. Part A is the first distance, part B is the second distance and part D is the distance from the ranch building.
ex2


This sketch is good to see the big picture of what is happening. Now we can take each part of the trip individually. Let's start with Part A.

Part A

Since the rancher moved due east, there is no trigonometry to do. Right before he turns, he is 1 km from the ranch building. Let's now deal with the second part of the trip.

Part B

Diagram 5 shows the right triangle from where the rancher turned to where he stopped.


Diagram 5. The second leg of the trip.
ex2a


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