Sunday, 6 December 2015

Trigonometric Graphs and The Unit Circle

After being introduced to the first 90 degrees of the trigonometric functions my students usually have to wait two or three years before I will let on that there is more. If you enjoyed my recent post 'Introducing Trigonometry' then you may be relived to know that you don't have to wait that long to see how I do it! 

The first thing I do is rewind a few years and review the concept of trig ratios using this sketch; asking the same sort of questions as I did in my last post. Now to avoid visual clutter I try to keep my GeoGebra sketches as stripped back as possible but you may have noticed that in this sketch, on one of the triangle vertex's I left the coordinate point visible - there was a reason for this. One of the difficulties that students have with understanding the unit circle approach is that one minute they're used to applying the trig functions to right-angled triangles of all sizes and orientations and the next thing they're looking at a special-case of a triangle constrained in a unit circle. They sometimes struggle to understand how the results we are deriving could in fact be derived from any size of circle. Obviously it's easy to prove the results can be generalised algebraically but to help ease them into the circle here's what I do. I click on the vertex I just mentioned and stick a trace on it by right-clicking and going down to trace on.

I also make sure I  have a trace on the ratios in the right-hand panel. As you slide theta you should see a quarter-circle being traced out - I'll explain that the way I set this sketch up was by making the hypotenuse constant and thus forcing the opposite and adjacent side lengths to change as theta changes. If you think about it only two side lengths need to change to allow theta to change. If you now change the triangle's size a little and repeat then your students should notice that the graph has not changed - this is simply to re-emphasise the fact that trig ratios are independent of triangle size.

"Now I'm going to propose something pretty off the wall - what do you think will happen if we allow this point to continue on its journey around a full circle? Any ideas how the graphs may look...well lets investigate"

Since the trig functions are independent of the size of triangle I will suggest that we set the size of the hypotenuse to one unit for convenience - at this stage we're ready for the journey around the unit circle.

The unit circle is obviously nothing new and I expect that most teachers already use it for teaching trigonometric graphs. You may not however have a dynamic visual to use with your students and I hope this sketch makes it nice and clear. If you are new to the unit circle then the basic idea is this. The fact that the hypotenuse is one unit means that the adjacent side length is simply $1\times Cos\theta$ and the opposite length is $1\times Sin\theta$. Thus the triangle's top vertex's horizontal displacement is equal to $Cos\theta$ from the origin and the vertical displacement to $Sin\theta$. Obviously I get my students to explain all of this to me...

"Why is the red line labeled with $Cos\theta$ and the blue line $Sin\theta$"

"What will the length of the  blue line be when $\theta=90°$ ?"

"What will happen to the length of the blue line once theta passes 90°?"

Meanwhile I will will check the $Sin\theta$ box to trace out the function as I slide theta and test their predictions. I will then generally get students to make some longer range predictions on mini-whiteboards. They can then test their predictions using the sketch up to 720°, forgive me for not allowing you to go on  forever but you can at least wind it back down to -180°. As I mentioned in my last post if you want plot rather than just trace out these functions you need to type in $y=Sin(x^o)$ into the input bar to make sure your plots are in degrees - press Alt+ o (as in the small-case letter) to get the degree symbol.

When you have spun through the sine and cosine functions you can turn to the tangent function. The question I like to ask is:

"Where does $Tan\theta$ show up on the diagram"

I can't blame them really - its seems as if it's just too irresistible but someone will inevitable suggest without thinking it through that its the hypotenuse. Not the right answer but with further consideration their misconception leads them to the discovery of the identity:

$Sin^2\theta+Cos^2\theta \equiv1$

Getting back to the idea that $Tan\theta$ is the ratio $\frac{opposite}{adjacent}$ brings us to a second identity all within the space of about a minute - I guess trig identities are like buses!

$Tan\theta  \equiv \frac{Sin\theta}{Cos\theta}$

I love the fact that you could spend a life time doing standard textbook trig problems and never notice these relationships but when you're shown the unit circle they are just staring you in the face!

Now we know what ratio to plot we can trace out the tangent graph and discuss why it's such an awkward customer compared to the smooth talking sine and cosine curves.

Try taking it for a spin!

P.S. An obvious topic that links with trigonometric graphs is graph transformations. You might like to check out my posts on 'Single Graph Transformations' and 'Successive Transformations' to help.

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