First off, wow, that unit circle post was by far the most viewed I’ve had. So maybe people would also be interested in this too.

The thing that’s difficult about graphing trig functions for the first time is that regular old graph paper just isn’t up to the job. When we have points like and to plot, you need something that is suited to the task, and the numbers involved. I have met teachers that say they just have students plot the multiples of , and then tell the students to trust them, it’s not really a zig-zag. And it seems to me that having them spend time plotting points in order to develop a *wrong* intuition is an amazingly bad use of time. You could just use the calculator and it would be better. But in order to make the time spent plotting points valuable you have to be able to see the shape of the curve from the points are plotted. So you need *y*-values of and and *x*-values that are multiples of and .

So behold:

Of course, you need a key for this:

You also need to call the students’ attention to how the *x*-gridlines aren’t evenly spaced, and that the unlabeled ones are multiples of and . But after that…

you have something that can allow student to connect their knowledge of the trig functions in the context of the unit circle to the traditional graphing context.

I don’t feel like wrestling with the code tags, so you can check out the latex source here, at sharelatex. (You don’t need a ShareLaTeX account to access that link, but if you’d like to sign up for one, please use this referral link, which lets me earn points toward referral bonuses. Thanks!) That document has two types of graphs– one for sine and cosine, and another for tangent and cotangent. There’s a second set with the functions graphed as well, which I don’t give to students, but it’s useful to have to point at in later classes.

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I try to get them to learn to set up their graph paper properly. Six boxes to pi/2, so that pi/6 is 2 boxes and pi/4 is 3 boxes. Four boxes up to 1, so that 1 is about equal to pi/3. I hope that drawing it in proportion will help them see the proportions better. But… maybe using your graph paper the first time will give them more support for starting to see. I’ll try it.

Yeah, it’s really a matter of how much new stuff do you want to hit them with at once. Making the transition from the unit circle— where the input is , is a point on the circle, and is the output of — to the traditional graphing on the plane is pretty huge. Minimizing the other things that they have to think about at that time is really helpful. Once they get a basic sense for how that works, I have them do their own for all the transformed functions.

Also, I should probably mention that I don’t use the same scale in the x- and y-directions on these. If you want to, though, it’s easy to change. Just change where I have [xscale=1.26, yscale=2] either to be the same number, or ditch the separate ones, and replace with something like [scale=1.5].