Trigonometry from First Principles

By Kemp, Andy | Mathematics Teaching, September 2009 | Go to article overview

Trigonometry from First Principles


Kemp, Andy, Mathematics Teaching


In the example Andy Kemp explores the roots of trigonometry.

(ProQuest: ... denotes formula omitted.)

I remember studying Trigonometry at school and wondering why it worked. It seemed, in some sense, magical that if I knew an angle and a side I could type something into my calculator and out would come the answer. For many years, and throughout my own education, it seemed sufficient to me that for some reason there was a function which related angles to sides in right-angled triangles. During mv ö ö ö Ö OJ ?-levels and my degree I added more layers of understanding, the concept of the unit circle, how to apply these functions to non-right-angled triangles and the relationships between these functions. But despite all of these I could never really see where Trigonometry came from and why in its basic form it only worked with right-angled triangles.

So one day when I came to teaching Trigonometry, I decided enough was enough and I was going to get my head round how it works, and design a way of introducing it to students that would explain some of the magic behind the Trigonometry.

I started from that central topic of the KS3KS4 course of proportion and similar shapes. Starting with a series of pre-printed triangles (see Worksheet 1), I asked the students to examine a collection of triangles and identify what was the same about some of the triangles. On the worksheet were nine triangles some of which were similar to each other. The students were encouraged to measure all the sides and all the angles then identify which ones were in some sense the 'same'.

At this point I am really hoping that the students will pick up on 'mathematically similar' as their understanding of the same, although I do allow them to pursue other definitions to begin with. Once they have settled on similar triangles, they are then asked to define some function which 'gives the same value for the triangles they think are the same, and a different value for those they think are different'. This step usually takes a while and students tend to need a lot of encouragement to keep trying options. I normally end up giving them 'hints' like try using the side lengths, or telling them they just need the normal operations + - × ÷. After a little guidance they usually spot that the ratio between two of the sides is the same for the similar shapes. With a little more guidance the students usually spot that actually there are six of these ratios which can be expressed as a proportion thus giving us a single number. When the triangles are similar these numbers are similar, but there is some confusion arising from how you label the sides.

The second worksheet gives students an opportunity to see how they can use the ratio between two of the sides, together with Pythagoras, to find the lengths of all the sides of a collection of rightangled triangles. The purpose of this exercise is to give students some idea of why it would be useful to be able to identify these ratios that we found for the previous worksheet, and begins to hint at the value of trigonometry.

However, at this stage there are stiH issues to do with how we name the sides, having up to this point used the labels of 'top', 'bottom' and 'hypotenuse'. This labelling system has problems as the values change if you rotate the triangle! At this point I ask the students to think about how we could label the triangle in terms of one of the angles to remove any ambiguity. I also tend to spend some time at this stage discussing why knowing one angle uniquely identifies our triangle - because the right angle means the other angle is predetermined. …

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