Monday, June 11, 2012

It's trigonometry Jim, but not as we know it!

Do you remember high school trigonometry? Was it a blurred sequence of formulae with a recipe book of incantations for solving standard exam questions? I'm afraid to say that was my experience and it has taken many years (decades!) for me to see the full beauty and unity of the subject. Teaching it for the first time this year I was determined not to inflict the same on my students. My attempt at something a little different was based on a few key metaphors and heavy use of diagrams and graphs, supported by stories to help students see why trigonometry is indeed an interesting study for scientists, mathematicians and historians.

Forget triangles - let's cast a horoscope!
In senior school trigonometry, we're not just playing with triangles any more. Welcome to the weird and wonderful world of circles and periodic functions. I think a good way to introduce the unit circle is to place it in the historical context of our ancient star gazers watching the heavens. That's our unit circle: the night sky. I suspect our ideas of angles, triangles and projections onto a circle owe as much to calculating the pharaoh's horoscope as they do to measuring irregular sizes strips of land to work out the taxes owed him.

Trigonometry : was it really about casting the daily horoscope?
Celestial dome cartoon (inset) from  

Now that we have the idea of the rotating angle, and students see (or better yet explore) the sine curve, I think it's well worth showing why this is such a relevant and powerful idea. Share with students some of the interesting physics - show wave motion, show how different light frequencies relate to different colours. Hint at the mathematical treasures that await them: I showed some pictorial sequences introducing Fourier analysis. Trigonometry is about so much more than triangles - and it's relevant, interesting and surprising.

Round about and round about we go : the trigonometry carousel
A powerful metaphor for explaining the rotating angle and the periodicity of trigonometric function is the carousel:

Think of the rotations around the unit circle like riding a carnival carousel.
Carousel photo CC-BY-NC- SA
Carousel horse photo CC-BY-NC-SA 
This emphasises the periodicity of the rotation and allows extension to the idea of spinning faster, going from $\sin(x)$ to $\sin(3x)$, or slower, $\sin(\frac{x}{2})$, and then spinning wider to $5\sin(x)$.

Draw a diagram!
At this stage I believe it's worth spending quality time looking at the different graphs of $\sin(x), \sin(3x), \sin(\frac{x}{2}), 5\sin(x), 5\sin(3x)$. This helps visualise the functions and helps avoid the problems we seen when students start working with $\frac{\sin(3x)}{3}$.  Time also to bring out your function machine analogies and emphasise that $\sin()$ is function operator - not a multiplication of $\sin \times x$.

When it comes to solving trig equations, so many text books are filled with pages of algebra and barely a single diagram. Want to really understand the equation $4\sin(3x) = 1$ in the range $0^\circ \leq x \leq 180^\circ$ ? Draw the graph (sketch or use your favourite graphing tool):

Now we have many stories to tell! Why are there so many solutions? Because your students understand we have periodic functions (riding the carousel) and have seen the difference between the graphs of  $\sin{x}$ and $\sin{3x}$ the reason for all those solutions becomes less mysterious. And a quick sketch can help check the solutions. Drawing the graphs of trig equations also reinforces the difference between equations (sometimes true) and identities (always true) - another source of confusion when students start trigonometry at this level.

A helpful place to use graphs is when teaching the Auxiliary Angle transformation of $a\sin(x) + b\cos(x)$. Remarkably, very few textbooks show the graphs. I started with the graphs: looking at the unexpected result that when we add a $sin()$ curve to a $cos()$ curve, we get another periodic function - just with a different amplitude and a slight phase shift.

Yes - that's physics coming in - I share this language with the students - many of them are studying physics. Once you show this remarkable graph, actually working out the equations for the transform is straight forward and it makes sense - it's not just abstract symbolic manipulation.

Trigonometric Identities : Same person, different clothes
A helpful metaphor to distinguish identities from equations (why oh why do we do regularly use the same equals sign to mean different things?):

Same equation, different clothes.
Superman/Clark Kent graphic by Ian MsQuee

Later, when it comes to teaching the transformations, I add in the telephone box:

The half-angle t-transform helps Superman transform back into Clark Kent.
And Clark Kent is easier to pin down (solve) than Superman!

Trigonometry at this level really is a lot of fun - it brings together many different ideas and skills, producing some beautiful and unexpected results. And so many more wonderful surprises to come for those students who will later visit the world of complex numbers! Hopefully we can share that wonder with our students, so they don't just view trigonometry as a set of definitions, formulae and algebraic manipulations, but instead develop a strong intuitive feel for working with the circular functions.

And don't forget the graphs - or as a colleague repeatedly tells her class "DRAW IT!"

Some Teaching Resources

  • The Maths 300 "Trigonometry Walk" lesson (subscription required - but an internet search will find you some worksheets) is an outdoor exercise that helps students get a strong sense for the idea of projecting onto the unit circle.
  • James Tanton's whimsical Squine and Cosquine presentation explores what would happen if we used a unit square instead of a unit circle. Perhaps save after students have consolidated the knowledge?
  • Vi Hart's 13 minute video "What is up with noises?" is a wonderful exploration of the physics of sound, music and hearing. A little long to fit into a busy schedule, but good for a rainy end-of-term day.

1 comment:

  1. Trigonometry is a very important topic of mathematics.Scientific fields that make use of trigonometry include:
    acoustics, architecture, astronomy , cartography, civil engineering, geophysics, crystallography, electrical engineering, electronics, land surveying and geodesy, many physical sciences, mechanical engineering, machining, medical imaging , number theory, oceanography, optics, pharmacology, probability theory, seismology, statistics, and visual perception
    That these fields involve trigonometry does not mean knowledge of trigonometry is needed in order to learn anything about them. It does mean that some things in these fields cannot be understood without trigonometry.