Bearings in math are a vital part of solving problems. They are used to calculate angles and distances and can be found using various methods.
As a mathematics teacher with over 14 years experience in the classroom I show you tips and tricks to make bearings easier and common mistakes to avoid.
This article will discuss how to find bearings in math and give example problems that use trigonometry and geometry. Let’s get started!
What does bearing mean in math?
In mathematics a bearing is the angle between two points.
Imagine you are looking at the diagram from a birds-eye view.
Type of bearings in mathematics
There are two types of bearings: true bearings and compass bearings.
True bearings measure angles in a clockwise direction from north while compass bearings use the compass points.
True bearings are given as three figures in degrees.
Compass bearings use north, south, east, and west.
Where are bearings used?
Bearings are used in navigation and surveying, as well as many other applications that require the measurement of angles.
In mathematics, bearings are used in Trigonometry.
How to calculate bearings in maths
The most common way to find bearings is by using trigonometry.
You can also use geometry including angles on parallel lines to find corresponding or alternate angles, then use the fact that angles at a point add to 360 degrees to help calculate bearings.
Knowing that the angle between North and East etc is 90 degrees is also helpful to find complementary angles.
A bearing example problem
Brett leaves home and runs north for 6 km. He then turns and runs east for 3km. What is the bearing of his current location from his home, to the nearest degree?
- First I would draw a diagram.
The bearing needed is the angle marked with the blue arc.
It is clockwise from north.
2. Use trigonometry to write an equation and solve it for θ, the bearing.
3km is opposite θ.
6km is adjacent to θ.
So the trig ratio we need is tan
How do you find a bearing between two points?
To find the bearing between two points it makes a difference which point you need to find the bearing from.
- Draw a north arrow on both points.
- Starting at the point where the question says ‘from’ go clockwise until you hit the line connecting the other point. This angle is the true bearing
Bearing between two points example problem 1
Find the bearing of X from Y.
Starting at point Y and turning clockwise until we hit the line connecting X, the bearing is the red arc.
Using geometry and the fact that there is 360 degrees in a revolution we can calculate that the bearing of X from Y is 360-120 = 240 degrees.
Bearing between 2 points example problem 2
What is the bearing of Y from X?
- First draw a north arrow on X.
- Notice that the two north arrows are parallel so we can use our knowledge of angles on parallel lines.
Our bearing is the arc in red in the diagram above and it is co-interior to the angle marked 120 degrees.
Co-interior angles on parallel lines add up to 180 degrees.
So our bearing of Y from X is 180-120 = 60 degrees.
Since we usually write true bearings as three digits we would say it is 060°T.
How to work out true bearings
To convert between a compass bearing and a true bearing first draw a diagram to see what quadrant the angle is in.
Use the fact that east is 90° clockwise from north, south is 180° from north and west is 270° clockwise from north or 90° anti-clockwise from north.
Then add or subtract to find the true bearing depending which compass bearing you are measuring it from.
For example, a compass bearing of S20°E is a true bearing of 160° T.
This is because south is 180° from north so you subtract 20° from 180°.
Here are some examples to convert a compass bearing to a true bearing.
|Compass bearing||Calculation||True bearing|
|N45°E||0° + 45°||045°T|
|S60°E||180° – 60°||120°T|
|S30°W||180° + 30°||210°T|
|N10°W||360° – 10°||350°T|
How do you solve a bearing problem?
Sometimes you may need to use non-right-angled trigonometry to help you solve a bearing problem.
In this case you need to know the sine rule and cosine rule.
Always start with a diagram if one isn’t given!
Using Trigonometry to find a bearing example problem
A boat is sinking 2 km out to sea from a marina. Its bearing is 062° from the marina and 342° from a rescue boat. The rescue boat is due east of the marina.
How far, correct to 2 decimal places, is the rescue boat from the sinking boat?
1. The first step is to find any other angles in the diagram.
Use the fact that the north arrows are parallel and that the rescue boat is due east, allows us to find the missing angles.
2. FInd the distance required. The side length in blue marked x is the distance we need to find.
Using the sine rule we get this equation:
And we can solve by multiplying both sides of the equation by sin 28.
Using Trigonometry to find a bearing example problem 2
Two planes leave an airport at the same time. The first travels on a bearing of 053° at 400 mph. The second travels on a bearing of 143° at 300 mph.
a) How far apart are the planes after 2 hours (correct to the nearest mile)?
b) Calculate, correct to the nearest degree, the bearing of the first plane from the second plane.
- First draw a diagram. Watch me draw the diagram and complete the solution to this problem on this video.
- You also need to find the distance each plane has travelled using distance = speed x time
- Use pythagoras’ theorem to find the distance between the planes
- For part b, draw true north arrows on each plane’s current location. Continue the north arrow south on plane 1
- Label the angle inside the triangle at the top right as θ
- Find θ.
- Find the angle between south and the line connecting the two planes.
- Using your knowledge of geometry including alternate angles on parallel lines to find the bearing required.
Bearings downloadable worksheet
Practice finding bearings on this free printable how to find bearings worksheet.
It includes answers as well as example problem 2 from above using trigonometry so you can have a printed copy of the question.
How to find bearings video
This video goes through the worked solutions from the bearings problems on the printable bearings worksheet.
The importance of mastering bearings in math
Bearings can be used in a variety of ways and are a key part of many mathematical problems. By understanding how to find bearings, you will be able to solve these problems with ease.
Bearings are also used in navigation and surveying, so it is important to master this skill if you plan on working in these fields.
Extra tips and tricks to make bearings easier
- Always draw a diagram if one isn’t given.
- Draw true north arrows on every point. These lines are parallel so you can now use angles on parallel lines to help you find unknown angles and bearings.
Common misconceptions and errors when finding bearings
- Not drawing north arrows on both points. This may result in you not seeing angles on parallel lines like co-interior angles and alternate angles
- Not finding all missing angles. This may mean you struggle to find the bearing or distance required if you are missing information for an equation.
- Not using the correct trig formula. You must learn SOHCAHTOA and be able to identify opposite and adjacent sides and the hypotenuse as well as non-right angled trigonometry formulas like the sine rule and cosine rule.
- Substituting incorrectly into the formula. Take care substituting correctly for each known angle and side.
- Not giving a true bearing as 3 digits. For angles less than 90° use a zero as the first digit.
- Using the anti-clockwise angle as the bearing instead of the clock-wise angle.
Final Thoughts on How to find Bearings in math
I hope you have found this article helpful in understanding how to use diagrams and angles when finding bearings in math.
Always remember to draw a diagram, and be sure to label all points and lines! If you have any questions about using geometry or trigonometry please don’t hesitate to reach out for help.
Your teacher is your first point of call but sometimes you may need help from an online math tutor for homework help or test prep. Here is my go-to Trigonometry book for more practise.