Trig Pictures, Part III

We almost have the whole picture in view.

Here’s where we left off:


In this diagram, the secant line segment is adjacent to the central angle θ.  And the tangent line segment is opposite that same angle.

But what if we want to draw our secants and tangents relative to the central angle 90-θ, the complement?  Let’s add those segments to the diagram and call them the cosecant and the cotangent.



Find similar triangles and then use ratios to establish each of the following:


(If you need help getting started, I did the first two of these for you in the last post.)


Find right triangles and then apply the Pythagorean theorem to establish each of the following:


Those two exercises reveal a lot of information all derived from that one diagram.  Once again…it’s better with pictures.

Picturing the Trigonometric Ratios: Step-by-step


Let’s start with the unit circle and a central angle θ that meets an arc on the circle:


The blue segment is called the “sine” of the angle.  It gets its name from a word that means “bowstring”.  Here is a diagram of a bow string truss.


So if you think of the intercepted arc as the bow and focus only on the upper half…


… you can see how they first decided to call that segment the “bowstring.”

But you do not have to hold your bow vertically.  If you use the complementary angle, 90-θ, you get the horizontal segment I have drawn in red (and re-drawn on the x-axis as well…you’ll see why soon.)


We can call that red segment the “bowstring of the complement”, or the “complement’s sine.”  Or we could just call it the cosine!

Now we are going to add two segments to the diagram: one that just touches the circle and one that crosses the circle.  We will use the same names we used for these back in geometry class.  As you can see from the labels, the orange one is called the tangent and the green one is the secant.


Now, as we examine this diagram, we can find a number of interesting pairs of similar triangles.  To discuss them, it would be helpful if we label some points of interest:


Now we have arrived at the key step.

Take a moment to convince yourself that ΔEBC is similar to ΔBDC. 

Then, since similar triangles have proportional sides, it must also be true that:


These are NOT arbitrarily chosen definitions.  They follow by necessity from the geometric definitions of these terms.

I will pause here to allow everyone to catch their breath.  In the next post, we will finish developing the diagram and add in the Pythagorean identities.

Also Better With Pictures — The Trigonometry Ratios

One of the digressions we follow almost every time I teach AP Physics is intended to give my students a better visual understanding of the 6 basic trig ratios.  But it also resolves several mysteries left over from their first introduction to trig.

When you first learned the trig ratios back in geometry class, you were probably taught the mnemonic SOHCAHTOA.  You can also find this many places on the web.  For example, here is a summary from


Already, we have a bit of a mystery.  Students don’t know where the word “sine” comes from, but that’s fine.  Lots of new topics introduce new vocabulary without going into the etymology of every new term.  But what about “cosine”?  What is co- about it?  A minor mystery to be sure, but even so…

Then we tell them about “tangent”.  We say it is “opposite over adjacent”, which is the TOA in SOHCAHTOA.  And a tiny bit of algebra shows that the tangent can also be written as sine over cosine.  But we still have a mystery:

What is “tangent” about the tangent? 

This is not just an etymology question.  The word “tangent” is already part of our math vocabulary.  We saw in geometry that at tangent line is a line that touches but does not cross a circle.  There are all kinds of rules about the angles and the lengths associated with tangent lines (and secant lines too, for that matter).  But what does that use of the word have to do with the way we are using it now, here in trigonometry?

From there, we move on to the reciprocal trig ratios:


And now you have more mysteries:

What does this secant have to do with the secant we learned about in geometry?

In geometry class, a secant is a line that crosses a circle.  Is there some reason we are re-purposing this vocabulary?

And as long as we are asking questions:

Why did you match “secant” with “one over COSINE”? And “cosecant”  with “one over SINE”?  Isn’t that making things unnecessarily complicated?

And once again:

What is “co-” about the cosecant?  And the cotangent?

All of these mysteries can be resolved with a single diagram.  I will develop the diagram step-by-step in my next post.  But for now, here it is in the finished form.  Ponder it well…


As you will see in the next posts, there are lots of similar triangles waiting to be discovered in this diagram.  Also, the Pythagorean theorem makes several useful appearances.

But for now, let’s resolve one of the mysteries:

Look at the segments labeled sine, secant and tangent.  Note their positions relative to the central angle, θ.

Now look at the segments labeled cosine, cosecant and cotangent.  Note their positions relative to the central angle 90-θ.  Each of these segments relate to 90-θ the way their corresponding segments relate to θ. And remember that 90-θ is called the “complement” of the angle.  So that’s where these segments get their names:

“the complement’s sine” —> cosine

“the complement’s secant” —> cosecant

“the complement’s tangent” —> cotangent

One mystery solved: “co-” is for “complement” !

More mysteries to be unraveled soon.


Those Flipping Inverse Graphs

"Sometimes the sophisticated thing is easier to understand 
than the kludgey thing. The flipped plots in the books were 
driving me crazy. They don't even pass the dimensional analysis 
 -- John Denker

I have been known to rant occasionally about some of the curriculum choices we have made for high school mathematics.  I feel the math we teach should be useful or interesting, and ideally both.  But it should also be true!  I think of that as the teacher’s equivalent of medical rule: first, do no harm.

What brings this to mind is an email discussion I have been having with Dr. John Denker, a physicist who I have “met” through the email list.  He has, as you see above, some very reasonable objections to the way that my book repeatedly points out the symmetry between the graph of a function and its inverse.  I would like to discuss his objections here in this post and also to show his suggested alternative. Then, in the spirit of this blog, our email discussion led me to remember a nice method for using graphs to evaluate iterated functions.  I will write about that in another post soon.

[By the way, to any current or future physics teachers out there: I strongly recommend joining the email list Phys-L.  It may be the single most helpful resource I know of, populated by many patient and enthusiastic experts, including Dr. Denker.  My current students can attest to the many occasions when my teaching has been assisted and influenced by ideas I encountered on Phys-L.]

OK, on to those flipping graphs…

CLAIM:  If you graph a function and its inverse on the same axes, the two graphs will have an interesting and esthetically pleasing symmetry with respect to the line y = x.

I am by no means the only one who makes this claim!  Search the web for “inverse function graph symmetry” and you will find that claim stated quite frequently.

For example, consider the squaring function, with its domain restricted to the non-negative real numbers.  It has an inverse function, the square root function, and sure enough, if you graph both functions on the same axes, you see the symmetry that I am talking about:


The graph above was made at, a very user-friendly graphing site.  And the graph certainly provides evidence in favor of the claim.  And I personally find this to be, as I said, esthetically pleasing.  Also, this idea is not without application.  When you learn a formula for finding derivatives of inverse functions, this symmetry will present a graphical way to understand that formula.  Still, if I had to guess, I suspect that most of the people who write about this claim do it because they like how the graphs look.  They look nice.


“If you graph a function and its inverse…”  But why would you ever do that?  As JD pointed out to me:

I don't think anybody ever needs to flip the axis to form 
the inverse function. Just read the axes in the other order.

That’s a good point.  For example, let’s look again at the squaring function by itself this time.


The graph is a convenient way to represent the relationship between the length of the side of a square and its corresponding area.  But if you already knew the area and wanted the length, you would not need to draw a new graph!  You would just need to know how to read the graph you already have!  You’d start on the vertical, area axis, go over to the curve and then read down to the horizontal axis where you would read the length.  Same graph, different purpose.



OK, I don’t need to graph the inverse function.  But what if I just want to because I like how it looks.  Is it wrong?  Does it violate the teacher’s version of the Hippocratic Oath?  Alas, not always, but often, and in a way that I should have considered, seeing as a major theme in my book is how math is actually used in science.  My current students may find it amusing that what I have overlooked are the units!  I may feel like graphing the two functions ON THE SAME AXES, but in most cases, it would be wrong.  Which brings us to…


The flipped graphs do not have the right units on each axis!

You don’t need to use sophisticated physics or chemistry to show this.  Our first graph above with the squaring function and the square root function illustrates the problem sufficiently.  Look again at the vertical axis.  Pick a number.  OK, 2.  But 2 WHAT? Is it inches or square inches? Meters or square meters?  D’oh.

The bottom line is that if your functions represent physical quantities, then those quantities have units.  And unless the units happen to be the same, then you cannot just flip the graph.  You have to flip the axes as well. You just can’t throw those graphs on the same axes.  Look at them side-by-side:


If you want to show the symmetry of the original claim, you could take these two graphs, put them on overhead transparencies and lay one on top of the other:


It’s not as pretty, but at least its not wrong.

When you are focusing on the math, it is easy to forget that these quantities have physical meaning and units that go with them.  For example, there are many books and websites where you can see a function and its derivative graphed on the same axes.  (It’s even done that way on some of the websites that I used in earlier posts about derivatives.)  But that makes no sense!  For example, how can you graph position and velocity on the same axes?  They don’t have the same units.

I did a quick web search for “function derivative graph applet”.  Here is one site (of many) I found that puts the function and derivative on the same axes:


Here’s one that correctly puts them on different axes:


These are both very clever websites, good for developing a feel for what a function’s derivative looks like.  But the second one is better — it remembers that these are quantities with units.  (Also, the shading on the derivative graph is a neat bit of fore-shadowing.  That’s a topic for another upcoming post as well.)

Getting back to the Flipping Graphs…

If you don’t like the looks of the two overlaid transparencies, there are some other ways to visualize these functions and their inverses.  John Denker presents a few of them on his website here:

Here is one diagram (figure 10) from his site that I especially liked:


The “reflector panel” reminded me of another clever method of interpreting graphs.  More on that in a future post…