It Stands to Reason
For the new GCSE pupils are expected to
- know the exact values of sin θ and cos θ for θ = 0º, 30º, 45º, 60º and 90º
- know the exact value of tan θ for θ = 0º, 30º, 45º and 60º
In the programmes of study the text is bold and underlined, so this is intended for the highest attaining pupils, for whom the factual recall and predictable exam questions that this material entails will not be inspiring. But taken as a context for inquiry, exploration and proof, this is a topic that can be rich and satisfying. @Letsgetmathing suggests starting with an activity in which pupils lean a 1m rule against a wall then measure the height of the triangle created, and hence calculate the sine of the angle for a range of different angles. They can estimate the values of sin θ in the special cases, and can also plot the values on a graph. This will suggest that sin 45º is about 0.7; they should ask if it’s exactly 0.7, and if it’s not, then what is it?
Start with an isosceles right-angled triangle (the short sides could be length 1, but better would be length x) on the board and ask what can be worked out: fingers crossed that “the length of the hypotenuse” and “the other two angles” are suggested quickly! The ratio opposite: hypotenuse (or, better, the scale factor from the length of the hypotenuse to the length of the opposite) is clearly , as is the scale factor from the length of the hypotenuse to the length of the adjacent (or, the ratio adjacent: hypotenuse); thus we know exactly the values of sin 45º and cos 45º. That the scale factor from the length of the adjacent to the length of the opposite is 1 tells us the exact value of tan 45º.
Once pupils have seen the idea that a well-chosen triangle can tell them exact values of trigonometric scale factors, the suggestion of using an equilateral triangle to determine, say, sin 60º should be quickly forthcoming from one pupil … and should be quickly challenged by another because it’s not a right-angled triangle … and a third pupil will suggest the idea of bisecting it … and a fourth will use Pythagoras to work out the height of the triangle … well, that’s the plan!
From this diagram the relevant scale factors come out naturally. It’s well worth then asking pupils if they notice relationships between the surd expressions. They’re likely to spot that cos 30º = sin 60º and vice versa, and this should be explained (by considering how the opposite and the adjacent swap, as it were, relative to the 30º and 60º angles), and then generalised to show that in any right angled triangle cos (90 – A) = sin A. Then they might consider, say, sin 30º ÷ cos 30º, and what they observe should be cross-checked with sin 60º ÷ cos 60º and sin 45º ÷ cos 45º; these then can be generalised to any right-angled triangle. It’s a bit of a leap to consider, say, sin230º ÷ cos230º; the hint to do so is in the square-root form of the surd expression. Again, what’s found here should be cross-checked with 60º and 45º, and then could be generalised.
Evaluating sin 0º and sin 90º (and the cosine scale factor as well) is a good introduction to the idea of a limit: if pupils consider a very, very tall right-angled triangle, they should be able to see that the side opposite the nearly-90º angle is nearly the same length as the hypotenuse, and thus the sine scale factor is nearly 1 and it gets closer to that value as the angle gets larger – we say it tends towards the limit 1; similarly, the cosine scale factor of the nearly-90º angle will be nearly 0, and will carry on getting smaller and smaller, tending towards the limit 0. A similar limit argument should be developed to explain why we say that sin 0º = 0 and cos 0º = 1
It’s important that our pupils know why these trigonometric scale factors have these values – but they do also need to remember the values as well. There are lots of mnemonics for doing so; this one (from @solvemymaths) will help them have the knowledge at their fingertips (comedy drum roll optional):
This from Great Maths Teaching Ideas is neat:
or you might like Mr Barton’s quirky left-hand finger methods.
Resourceaholic contains three good suggestions to challenge pupils’ knowledge and retention of the scale factor values. The code breaking task is especially fun.
Making Maths: a Clinometer from NRICH is a lovely activity for exploring trigonometry, using real-life heights of wind turbines and trees.
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