## Trigonometry

After school this evening I checked my twitter feed and was met by this confusing tweet:

After some investigation it seems that the tweeter Ben Rouse (@Mr_BRouse) was engaging in a process of “pimping” some trig questions to make them “interesting”. Having seen a few of the exchanges after, and having looked at his links, I think he means add more context/make more practical. The questions he posted were specifically what I call “RAT Trig” (Right Angled Triangle Trigonometry), and didn’t include Pythagoras’s Theorem.

I can understand the desire to contextualise these questions, and show pupils why they might use this. I often use real life type questions teaching this. Infact I planned a lesson around it with the help of a friend of mine who is a joiner, and have noted the cross over with the construction teacher at school. I’m sure Ben’s lesson will be great and I look forward to hearing more about it and seeing the “Pimped” questions it produces. Especially if they are all as good as Oli Trussell’s (@olivertrussell) offerings!

However, the phrase struck me as odd, I find triangles incredibly fascinating shapes. I love the properties they present, the links they have. I find Trigonometry, on the whole, and incredibly interesting area if mathematics. From RAT Trig all the way through to de Moivres Theorem and complex analysis. Further, into the hyperbolic functions, the links to e. Trigonometry is, in many ways, our basis to understand the world.

When I introduce Trigonometry I spend a lot of time investigating and discussing the properties of similar triangles, looking at how and why the trigonometric ratios work. Showing the link between the tangent ratio and the gradient of a straight line. The links to Pythagoras’s Theorem and coordinate geometry. When I move on to the sine and cosine rules I look at the links back. I investigate how Pythagoras’s Theorem is just a special case of the cosine rule, and how Sin = opp/hyp is just the sine rule.

I love the properties of triangles, and all their uses and I could investigate it ad infinitum. That’s why I tweeted back ‘but trigonometry IS interesting”, I was surprised to receive a reply from Dave Gale (@reflectivemaths) which read “No it isn’t”. I know Dave loves maths, I listen to his podcast and read his blog, so why can’t he seem the beauty and amazement that Trigonometry presents?! His normal stance, or go to argument against a pure maths idea is “But it’s not useful!” this, however, doesn’t fit here. Mechanics is based entirely on trig. And Mechanics is arguably the area of maths that has the most “real life” uses. Not only does it help us understand the world around us, it enables us to build solid structures and basically have a civilisation!

Pythagoras is most definitely part of trigonometry, in fact it is a special case. Consider the cosine rule for non-right angle triangles: c²=a²+b²-2ab.cosC

When side c is the hypotenuse of a right angle triangle, (the angle opposite) C = 90°,

so -2ab.cos(90) = 0 and hence you are left with Pythagoras’ Theorem c²=a²+b².

Aye, that’s what I meant by “I investigate how Pythagoras’s Theorem is a special case of the cosine rule”!

Yeah, I was just climbing onto that train and adding words to it 🙂

Reblogged this on The Echo Chamber.

You can also extropolate it out to say regular polygons are a combination of triangles as well (linking it to interior and exterior angles). Makes for some great problems

Aye, I’ve had some great fun with those types of problem.