Jour 18 : Quadratic functions and the ambiguous case of the law of sines

I have been spiralling the curriculum and the course content of MCR3U for 2 years now and I am always making new discoveries.  When one starts to think about relationships between mathematical concepts, one truly starts to think about the math behind the procedure/the rule/the formula…I have loved where spiralling 3U has taken me and I think this most recent discovery is among my favourites!

Here is how the thinking happenened :

How do I spiral solving triangles with trig ratios and laws into a curriculum that centers around functions?

As I was working on the solving triangles cycle, I started thinking about tangents and secants when looking at this representation of the ambiguous case of the law of sines :

Si une tangente et une sécante peuvent être définies en utilisant un cercle, je devrais pouvoir faire un lien avec les solutions des systèmes d’équations qui impliquent une fonction affine et une fonction quadratique.

Alors, ma première tentative avec ce cycle était au dernier semestre lorsque j’ai défini le mot tangente et sécante pour les élèves en utilisant le cas ambigu.  De là, nous avons exploré les points d’intersection entre une fonction affine et une fonction quadratique.

But this representation and solutions to a linear/quadratic system got me thinking!

Doesn’t the ambiguous case of the law of sines have 0, 1 and 2 solutions?

Doesn’t a linear/quadratic system have 0, 1 or 2 points of intersection?

Also, doesn’t a quadratic equation have 0, 1 or 2 x-intercepts?

And here’s the big moment

DOESN’T THE LAW OF COSINES LOOK LIKE A QUADRATIC EQUATION!!!

So, what if we tried to solve an ambiguous case of the law of sines (or any triangle for that matter) with the law of cosines instead of the law of sines.

Alors, j’ai demandé aux élèves de revisiter des triangles qu’ils ont résous avec la loi des sinus et de tenter une application avec la loi du cosinus au lieu.  Je me suis assuré de prendre au moins un exemple d’un cas ambigu, en plus des exemples que les élèves tentaient.

And this is what happened!

Maintenant, avec la formule quadratique, il s’agit de trouver les valeurs possibles de c. Puisque ceci était un cas avec 2 solutions, il y avait 2 valeurs possibles de c.

Grande idée pour les élèves :

Est-ce qu’on pourrait donc résoudre n’importe quel triangle avec simplement la loi du cosinus et une connaissance des fonctions quadratiques?

Now isn’t that beautiful???  When students can realize that there isn’t a magical way to solve a problem.  Triangle A doesn’t have to use SOHCAHTOA only and Triangle B doesn’t have to be solved using the law of sines…what if we explored solving all triangles for side or angle with different laws.  Then, the students can make their own rules about which one is more appropriate and at which time!

Let the students discover when one law of procedure fails, when one is successful and when one is quicker than another.

Now, my students have a choice.  They can keep applying the law of sines or they can use the law of cosines when faced with the ambiguous case, or any triangle for that matter.  This empowers them!

DISCLAIMER

Now, I’m not the first person to have seen this relationship.  You can see it in this video here that I found when I googled to see if anyone else had thought of this.