More Useful Trig Graphics

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 $\left(\frac{\pi}{4} , \frac{\sqrt2}{2} \right)$ and $\left(\frac{\pi}{3} , \frac{\sqrt3}{2} \right)$ 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 $\frac{\pi}{2}$, 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 $\pm\frac{\sqrt2}{2}$ and $\pm\frac{\sqrt3}{2}$ and x-values that are multiples of $\frac{\pi}{6}$ and $\frac{\pi}{4}$.

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 $\frac{\pi}{6}$ and $\frac{\pi}{4}$.  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.

TikZ Unit Circle

After getting several requests from IRL folks for my unit circle, I figured I’d make it a post here. (Seriously, when I get tackled in the hall for a unit circle handout, it means that there’s a real need out there, even though I would have figured that there’s enough unit circles in the world.)

Blank unit circle

blank unit circle as pdf

latex code on sharelatex.
latex code:

\documentclass[border=4pt]{standalone}
\usepackage{amsmath,mathpazo,gensymb}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture} [>=stealth, scale=2.5,
% Toggle commenting on the next four lines for the completed unit circle:
angle/.style={draw,text=white,fill=white,minimum height=1cm, minimum width=1cm},
point/.style={white},
%angle/.style={fill=white},
%point/.style={},
]
\draw [white] (-3.6,-3.6) rectangle (3.6,3.6);
\draw [thick,fill=white] (0,0) circle (3cm);
\draw [thick, ] (-3.3,0) -- (3.3,0);
\draw [thick, ] (0,-3.3) -- (0,3.3);
\draw (0,0) --  node [angle] {$\frac{\pi}{2}$} (90:3)
node[point, above right] {$\left(0,1\right)$};
\draw (0,0) --  node [angle] {\footnotesize{$\pi$}} (180:3)
node[point, above left] {$\left(-1,0\right)$};
\draw (0,0) --  node [angle] {$\frac{3\pi}{2}$} (270:3)
node[point, below right] {$\left(0,-1\right)$};
\draw (0,0) --  node [angle] {\footnotesize{$2\pi$}} (0:3)
node[point, above right] {$\left(1,0\right)$};
\draw (0,0) --  node [angle] {$\frac{\pi}{4}$} (45:3)
node [point, above right] {$\left( \frac{\sqrt2}{2} , \frac{\sqrt2}{2} \right)$};
\draw (0,0) --  node [angle] {$\frac{3\pi}{4}$} (135:3)
node [point, above left] {$\left( -\frac{\sqrt2}{2} , \frac{\sqrt2}{2} \right)$};
\draw (0,0) --  node [angle] {$\frac{5\pi}{4}$} (225:3)
node [point, below left] {$\left( -\frac{\sqrt2}{2} , -\frac{\sqrt2}{2} \right)$};
\draw (0,0) --  node [angle] {$\frac{7\pi}{4}$} (315:3)
node [point, below right] {$\left( \frac{\sqrt2}{2} , -\frac{\sqrt2}{2} \right)$};
\draw (0,0) --  node [near end, angle] {$\frac{\pi}{6}$} (30:3)
node [point, above right] {$\left( \frac{\sqrt3}{2} , \frac{1}{2} \right)$};
\draw (0,0) --  node [near end, angle] {$\frac{\pi}{3}$} (60:3)
node [point, above right] {$\left( \frac{1}{2} , \frac{\sqrt3}{2} \right)$};
\draw (0,0) --  node [near end, angle] {$\frac{2\pi}{3}$} (120:3)
node [point, above left] {$\left( -\frac{1}{2} , \frac{\sqrt3}{2} \right)$};
\draw (0,0) --  node [near end, angle] {$\frac{5\pi}{6}$} (150:3)
node [point, above left] {$\left(- \frac{\sqrt3}{2} , \frac{1}{2} \right)$};
\draw (0,0) --  node [near end, angle] {$\frac{7\pi}{6}$} (210:3)
node [point, below left] {$\left(- \frac{\sqrt3}{2} , -\frac{1}{2} \right)$};
\draw (0,0) --  node [near end, angle] {$\frac{4\pi}{3}$} (240:3)
node [point, below left] {$\left( -\frac{1}{2} , -\frac{\sqrt3}{2} \right)$};
\draw (0,0) --  node [near end, angle] {$\frac{5\pi}{3}$} (300:3)
node [point, below right] {$\left( \frac{1}{2} , -\frac{\sqrt3}{2} \right)$};
\draw (0,0) --  node [near end, angle] {$\frac{11\pi}{6}$} (330:3)
node [point, below right] {$\left( \frac{\sqrt3}{2} , -\frac{1}{2} \right)$};

\foreach \n in {1,2,3,4}
\foreach \t in {0,30,45,60}
\fill (\n*90+\t:3) circle (0.03cm);
\foreach \t in {30,45,60, 120,135,150, 210,225,240, 300,315,330}
\node [font=\tiny, fill=white,inner sep=1pt] at (\t:.75) {$\t\degree$};
\end{tikzpicture}

\end{document}


Completed Unit Circle

completed unit circle as pdf

latex code on sharelatex
Get the latex code here.

\documentclass[border=4pt]{standalone}
\usepackage{amsmath,mathpazo,gensymb}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture} [>=stealth, scale=2.5,
% Toggle commenting on the next four lines for the completed unit circle:
%angle/.style={draw,text=white,fill=white,minimum height=1cm, minimum width=1cm},
%point/.style={white},
angle/.style={fill=white},
point/.style={},
]
\draw [white] (-3.6,-3.6) rectangle (3.6,3.6);
\draw [thick,fill=white] (0,0) circle (3cm);
\draw [thick, ] (-3.3,0) -- (3.3,0);
\draw [thick, ] (0,-3.3) -- (0,3.3);
\draw (0,0) --  node [angle] {$\frac{\pi}{2}$} (90:3)
node[point, above right] {$\left(0,1\right)$};
\draw (0,0) --  node [angle] {\footnotesize{$\pi$}} (180:3)
node[point, above left] {$\left(-1,0\right)$};
\draw (0,0) --  node [angle] {$\frac{3\pi}{2}$} (270:3)
node[point, below right] {$\left(0,-1\right)$};
\draw (0,0) --  node [angle] {\footnotesize{$2\pi$}} (0:3)
node[point, above right] {$\left(1,0\right)$};
\draw (0,0) --  node [angle] {$\frac{\pi}{4}$} (45:3)
node [point, above right] {$\left( \frac{\sqrt2}{2} , \frac{\sqrt2}{2} \right)$};
\draw (0,0) --  node [angle] {$\frac{3\pi}{4}$} (135:3)
node [point, above left] {$\left( -\frac{\sqrt2}{2} , \frac{\sqrt2}{2} \right)$};
\draw (0,0) --  node [angle] {$\frac{5\pi}{4}$} (225:3)
node [point, below left] {$\left( -\frac{\sqrt2}{2} , -\frac{\sqrt2}{2} \right)$};
\draw (0,0) --  node [angle] {$\frac{7\pi}{4}$} (315:3)
node [point, below right] {$\left( \frac{\sqrt2}{2} , -\frac{\sqrt2}{2} \right)$};
\draw (0,0) --  node [near end, angle] {$\frac{\pi}{6}$} (30:3)
node [point, above right] {$\left( \frac{\sqrt3}{2} , \frac{1}{2} \right)$};
\draw (0,0) --  node [near end, angle] {$\frac{\pi}{3}$} (60:3)
node [point, above right] {$\left( \frac{1}{2} , \frac{\sqrt3}{2} \right)$};
\draw (0,0) --  node [near end, angle] {$\frac{2\pi}{3}$} (120:3)
node [point, above left] {$\left( -\frac{1}{2} , \frac{\sqrt3}{2} \right)$};
\draw (0,0) --  node [near end, angle] {$\frac{5\pi}{6}$} (150:3)
node [point, above left] {$\left(- \frac{\sqrt3}{2} , \frac{1}{2} \right)$};
\draw (0,0) --  node [near end, angle] {$\frac{7\pi}{6}$} (210:3)
node [point, below left] {$\left(- \frac{\sqrt3}{2} , -\frac{1}{2} \right)$};
\draw (0,0) --  node [near end, angle] {$\frac{4\pi}{3}$} (240:3)
node [point, below left] {$\left( -\frac{1}{2} , -\frac{\sqrt3}{2} \right)$};
\draw (0,0) --  node [near end, angle] {$\frac{5\pi}{3}$} (300:3)
node [point, below right] {$\left( \frac{1}{2} , -\frac{\sqrt3}{2} \right)$};
\draw (0,0) --  node [near end, angle] {$\frac{11\pi}{6}$} (330:3)
node [point, below right] {$\left( \frac{\sqrt3}{2} , -\frac{1}{2} \right)$};

\foreach \n in {1,2,3,4}
\foreach \t in {0,30,45,60}
\fill (\n*90+\t:3) circle (0.03cm);
%	\foreach \t in {30,45,60, 120,135,150, 210,225,240, 300,315,330}
%		\node [font=\tiny, fill=white,inner sep=1pt] at (\t:.75) {$\t\degree$};
\end{tikzpicture}

\end{document}


As a sidenote, I am really pleased that WordPress’s code tags seem to have improved greatly since the last time I tried to use them.  Never mind.  It’s better than it was, but it’s changing < and > to < and >.  And it took the away entirely.  For the actual code, you can check out the versions on sharelatex, as well as this document, that has them both embedded in it.  (You don’t need a ShareLaTeX account to access these links, but if you’d like to sign up for one, please use this referral link, which lets me earn points toward referral bonuses. Thanks!)

Altering the Mathematical Collaboration Expectations

One thing that needs to be added to my Mathematical Collaboration Expectations is perseverance.

I can’t believe that I didn’t even realize that I didn’t have that on there.  Now, is it its own category, or does it fit under something else?  I’m leaning towards its own category, but then I have to break it down.  What are the qualities of perseverance?