Revisiting the Cosine Rule

The cosine rule (called law of cosines in the USA) is taught in England for the GCSE (at the end of year 11, which is the equivalent to grade 10 in the US) while it is taught in France only in year 13. This realisation led me to analyse further how it is taught in England and also to explore further the cosine rule itself.

First, I noticed that the most commonly used formula for the cosine rule in England is

in the triangle ABC where AB = c, BC = a, and AC = b, and the vertices A, B, and C are confused with the angles CAB, ABC, and BCA. [Sometimes, the three formula are given, starting with this one and then having b²= c² + a²– 2 ca cos B and c² = a² + b²– 2 ab cos C.]

I am surprised that not c² = a² + b²– 2 ab cos C is used as main formula instead because it is easier with this formula to see that Pythagoras is only a special case of the cosine rule: when the angle is 90 degrees, then we find indeed that c² = a² + b² (because cos C = 0 when C = 90 degrees), which is the formula generally used for Pythagoras’ theorem.

GCSE students know the cosine of an angle as a side ratio in a right-angled triangle. Even if they have already learned about the cosine function that defines the cosine value of an angle for any angle value, they haven’t seen yet the extended geometrical definition of cosine in a unit circle. So, when sketching the graph of the cosine function, they will be able to say that cos 90 = 0, but they are probably not thinking of this case in a geometrical context.

When considering the cosine rule as a generalisation of Pythagoras’ theorem in any triangle, as I propose below, we see that it can be used to show that the cosines of two supplementary angles have the same absolute value but an opposite sign.

I propose an activity (for A-level students or top GCSE students) to revisit the cosine rule, making the link with Pythagoras’ theorem and extending the definition of the cosine of an angle in a right-angled triangle to the case of scalene triangles, seeing that the cosine of an acute angle (Part A) is positive, while that of an obtuse angle (Part B) is negative. Links between this geometric definition and the graph of the cosine function are then made (Part C).