Calculating with cosines
If you are a teacher, click here for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on...
If I tell you two sides of a right-angled triangle, you can easily work out the third using Pythagoras' theorem. But what if the angle between the two sides is not a right angle? Is there a way to work out the length of the third side?
Some students tried to work out a formula to work out the length of the third side when we know two sides and the angle between them.
Below, you can see the start of their methods. Can you finish each method to produce a formula?
Student 1:
Student 2:
Student 3:
Does each method lead to an equivalent formula?
Does each method work for both acute-angled and obtuse-angled triangles?
Try to make sense of each method shown one line at a time. The following mathematical topics may be useful:
Pythagoras' theorem
Trigonometry in right-angled triangles
Cosines of angles greater than $90^{\circ}$
Distance between two points on a coordinate grid
Student 1
Gautham from Kendriya Vidyalaya SAP Perookada TVPM in India, Sam from Bedford school, Joshua from Bohunt Sixth Form in the UK, Soumya from Kings Maths School in the UK and Nicholas completed Student 1's method. Here is Sam's work (click on the image to see a larger version):
Not everyone's formula came out exactly the same. Gautham left the formula in a different way (click on the image to see a larger version). Can you see how the two are equivalent?
Nicholas thought about whether this method would work for obtuse-angled triangles:
Our method only works when C is acute since our initial construction involved a right angle triangle involving C and the angles in a right angle triangle must be less than 90.
This method be adapted for triangles where C is an obtuse angle. Can you work out how?
Student 2
Sam, Joshua, Nicholas, Soumya and Gautham all completed this method. Once again, they didn't all express the formula in the same way. Soumya got the same formula as Gautham (click on the image to enlarge):
Joshua got the same formula as Sam:
Once again, Nicholas also considered what might happen if C were obtuse (click on the image to enlarge):
Student 3
Sam, Joshua, Nicholas, Soumya and Gautham all completed the method. This is Soumya's work, which shows the third diagram (click on the image to enlarge).
In this case, C is obtuse. Nicholas concluded that this must always be the case (click on the image to enlarge):
What would happen if all the angles were acute? Try out the method in that situation - it still works!
Why do this problem?
Many students are introduced to the cosine rule for the first time as a complicated formula they need to remember. In this problem, students are encouraged to seek a formula to work out the unknown side in a triangle. The suggested starting points can lead to deduction of the cosine rule, and by exploring it in several ways students will gain insight into where the formula comes from.
Possible approach
Here is a way to approach this task with students who have not met the cosine rule.
"If I gave you this triangle, I'm sure you could work out the length of the missing side:"
"Here's a slightly different triangle. Do you think the missing side is longer or shorter? How do you know?"
"Is there a way of calculating the missing side if we know two of the sides and the angle in between them?"
Give students some time to work in small groups on trying to calculate the missing side for the particular example above, and then if they succeed, trying to generalise. Some suggested prompts you might offer:
"Could you add any lines to the diagram to create right angles?"
"Once you've created a right-angled triangle, what dimensions do you know? What can you work out?"
"If you drew the triangle on a coordinate grid, what would the coordinates of the vertices be?"
"If you know two points on a coordinate grid, how can you work out the distance between them?"
Once students have had a go at working out the missing side (either for this example or for the general case), offer them this worksheet which contains three starting points for working out the cosine rule.
You may wish to split the class into groups and give each group one method, before bringing the class together to demonstrate their methods to each other. Alternatively, all groups could work on all three methods and the session could finish with students evaluating the three methods and discussing which they prefer.
Key questions
For Student 1's approach:
Can you make sense of each line of the method?
How can we express length x in terms of a, b and angle C?
Would the method work if angle C was obtuse?
For Student 2's approach:
Can you place any triangle on a coordinate grid in this way?
Would the method work if angle C was obtuse?
For Student 3's approach:
How can $\cos(180^{\circ}-C)$ be expressed in terms of $\cos C$?
Possible extension
Challenge students to find a formula to work out the area of a triangle given two sides and the angle between them. This can then be used to prove the Sine Rule $(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C})$.
For those methods which do not work for obtuse angled triangles students could be challenged to extend the argument so that they do.
Possible support
Student 1's and Student 2's approach are more straightforward so students could just work on those two. By working in small groups, students can support each other to make sense of each method.