Introducing Trigonometry

I love introducing trigonometry - it's one of those big new concepts to get excited about teaching. It simply blows my mind that a few simple relationships between the dimensions of a right-angled triangle underpin the mathematics of a vast array of diverse fields; astronomy, acoustics, optics, seismology,  electronics, civil engineering and mechanical engineering to name just a few. I feel it's kind of important to kick it off with a good start!

I have seen trig introduced a number of different ways ranging from the mega investigative approach outline by Jo Morgan in her awesome post on resourceaholic.com through to... well, it been indigently given no real introduction at all! :(

My approach tends to vary depending on the time available and the type of class but one thing that now always ends up in the mix is a little GeoGebra. I have found that in particular the sketch below is fantastic for developing students conceptual understanding of trig ratios in even a relatively short space of time. This makes it a particularly great tool for introducing the topic when time is short, for revision purposes, or to summarize students findings after a longer paper based investigation. This sketch can be used in many different ways but I will outline a common approach I use that dips in and out of it over the course of a few lessons.

First off I get everyone in the class to draw an accurate right angle triangle that is any size they like but has another given angle, lets say \(36^o\) - a nice bit of practise in constructing a RHS triangle. 

Next up I introduce the terminology of 'adjacent' and 'opposite' sides and get them to measure and record these lengths, then finally I'll get them to  calculate the opposite/adjacent ratio. I'll take some answers and put them on the board... 

"That's weird, you all seem to have roughly the same answer...is that weird?"

This usually provokes an interesting discussion where terms like 'similar triangles' and 'gradient' often feature. At this point I'll show them my GeoGebra construction - I'll set theta to \(36^o\) first and the click the lower check box to reveal the tangent ratio. Playing around with the size of the triangle reiterates what they have just found as a group - the ratio is independent of the size of the triangle.

A nice game is then to find out who drew and measured most accurately; this gets students sharpening their pencils and honing there protractor and ruler skills as they attempt to draw a second triangle with a different angle - this can get quite competitive. If you find it's a tie you may need to resort to ramping up the the number of decimal places displayed in GeoGebra (I explain how here).

Its then back to GeoGebra to check - I might go for a third triangle or maybe we'll try to figure out a couple of ratios:

"What will the ratio be when \(\theta = 0\)?"

"How about when when \(\theta = 90\)?"

"\(\theta = 45\)?"

At this point I'll use GeoGebra to demonstrate that each angle has a unique ratio and trace up \(y=Tan(\theta)\)  by checking the green trace button and sliding theta (I won't mention what happens outside of the range 0-90 at this point). Note that if you want to plot the actual function click anywhere in the right-hand panel, then in the 'input bar' at the bottom type \(y=Tan(x^o)\) - GeoGebra always uses x as the horizontal axis variable and you must include the degree symbol to ensure the plot is in degrees not radians. Hint: Press Alt+o (as in the small-case letter) to get the degree symbol.

Why is GeoGebra having trouble calculating the ratio?

I will then explain that you can access the ratio for any angle by looking it up in a table or more conveniently using the 'tan' button on your calculator! From here I'll develop the idea of how we can use the tangent ratio to find the adjacent if we know the opposite and visa versa and we'll tackle some typical textbook type problems using the tangent ratio.

Over the next couple of lessons we'll come back to the GeoGebra sketch to explore the Sin and Cos functions in a similar way, again emphasising the questions: 

"What will the ratio be when \(\theta = 0\)?"

"How about when when \(\theta = 90\)?"

to get students really thing about what is happening to the ratios as theta changes.

"Will the sine and cosine ratios ever be the same? When and why?"

"Do we actually need all three ratios to solve right angled triangle problems?"

"Why do the sine and cosine functions have some similarities in their graphs but the tangent graph is totally different"

This sketch also makes a great starting point for introducing the trigonometric graphs via the unit circle which I will discuss in a subsequent post. I hope this sketch is useful to you whichever way you introduce trig - thanks for reading.