Issues in math
education
Making Trigonometry More than
SOHCAHTOA
By Dan Woelders
Daniel Woelders is a math teacher and department head at Pacific Academy High School in
Surrey, BC. Over the past 5 years, he has taught in the Abbotsford and Langley school districts
in both a math and science classroom. He is currently in the process of completing his Masters
of Education in Secondary Mathematics at Simon Fraser University and is an active blogger on
the subject of mathematics and math education.
E
very year that I have taught high school mathematics I have had the opportunity to
introduce students to trigonometry. This is an opportunity show that trigonometry is
not only a tool to solve triangles but also a different way of describing the slope of a
line. Despite my best efforts, students seem to complete the course and carry forth a limited
understanding about trigonometric ratios. If asked what they understand about trigonometry
they consistently answer, “SOHCAHTOA” and when prompted to explain what that means
they reply, “Sine is opposite over adjacent”. No doubt this is a description of the sine ratio at a
very basic level but in no means does it touch at the very heart of its purpose. Rarely do you
hear a student describe sine as a relationship between an angle and the ratio of its opposite side
to the adjacent side of any right triangle. While the argument could be made that they knew
it but couldn’t explain it, this hardly seems likely given my follow up with students following
the unit. Some of the following questions have been posed to students previously and were
intended to get at their conceptual understanding of trigonometric ratios.
Could there be more than three ratios? Explain your reasoning.
Some of the more comedic responses include “because there is only three buttons on our
calculator” or “yes, there are more than three ratios, sin-1, cos-1, and tan-1”. However, the most
common response was there was only three ratios because there are only three sides or perhaps
that there were only three angles. These are true statements in a roundabout way but they miss
the idea that three sides can only be paired in three ways. If they saw the ratios in this way
they might have some insight as to why there can actually be six ratios if you were to flip all
three ratios upside down. Students might even argue why the sine ratio needs to be opposite/
hypotenuse instead of hypotenuse/opposite. This may lead to the discovery of the graph of
each of these functions.
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Does the sine ratio have a maximum value? If so, what is it? Explain your reasoning.
Responses to this question varied in the past. Often students responded incorrectly that the
sine ratio maxed out at 90 degrees because you can’t have a triangle with two 90 degree angles.
Part of this, of course, is true, but students misunderstood the nature of the increasing sine
ratio and often mistook it as a positive linear correlation. They assumed that the ratio increases
as the angle increases. This becomes more evident in the next question.
If the cosine ratio of 60 degrees is 0.5, estimate the cosine ratio of 30 degrees? Describe how you
arrived at your answer (this is done without the use of a calculator).
Not surprisingly most students tend to answer that the cosine ratio of 30 degrees would be
0.25. The same reason that students misunderstood maximum values for sine ratio resurfaces.
Once again the assumption is that trigonometric ratios and angles form some sort of linear
relation. This is potentially hazardous as student begin to interpret the periodic motion of the
sine and cosine functions.
Estimate the angle that would have a tangent ratio of 2? Describe how you arrived at your answer.
The response to this question varied drastically, some relatively accurate, others dipped below
45 degrees. The most frequent inaccurate response was 2 degrees, suggesting that either the
student had no tools to even venture a guess or they truly believe that ratio and angle were
equivalent. The misunderstandings extend beyond the evidence that I have accumulated in
post-unit surveys. It was not uncommon to see solutions to trigonometric statements that
make you question whether students really understand anything about the concepts involved
in trigonometry. The sin 60 = x/6 was often answered by stating the value of x in degrees (as
opposed to a length unit).
This has bothered me for years, but I often dismissed it as the student’s fault. Year after year I
taught trigonometry the same way, and year after year their grasp of the concepts was dismal.
“Students are slow”, I excused. Now I wonder who the slow one is. The approach to the unit
was a definition-based approach, filled with equations and algorithms, and the students were
able to do exactly that. They could solve a triangle and do the algebra. The following is an
example of a common approach I took to introducing trigonometry.
14
14
10
30
x
x
In the past we have been able to solve this triangle by using the Pythagoras equation. However,
the triangle right does not have two sides that we need to find the third side. Instead, it gives
us the angle. Furthermore, using Pythagoras doesn’t allow us to find the angles of the triangle
on the left. Now, we can use trigonometry to find this information.
Now, this is not to say it is a terrible approach. It necessitates the use of trigonometry and
would maybe accompany a real life example really well. However, this introduction was often
followed by stating one of the three ratios. Right away the students experience trigonometry
from a formula-based approach. It also begins with a foreign term like “cosine” and quickly
defines it as opposite/adjacent. To the students cosine is not a ratio, it is a formula for solving
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triangles. The first lesson consisted of getting them to memorize that tangent was opposite/
adjacent, cosine was adjacent/hypotenuse, and sine was opposite/hypotenuse. I would then
hand them a worksheet where they spent 30 minutes labelling triangles with the letters O,
A, H and theta (another strange adjustment in their understanding of variables that I rarely
explained). Historically the second lesson of the trigonometry unit really excited me. It was
the lesson where I would get to tell them what that button on their calculator labelled “sin”
did. “If you hit ‘sin’ followed by any angle you want, it will tell you the opposite/adjacent”, I
would explain. In my attempt to teach trigonometry I deprived students the opportunity to
experience the world of mathematics. In the book, “Developing Understanding of Geometry”
(Sinclair & Pimm, 2012)1 , the authors identify four big ideas that should drive our approach
to geometry. Using these big ideas, I will suggest the problems that emerge in the lesson
described above and introduce a new approach that I have recently experimented with and
the results of that approach.
Big Idea 1: Working with diagrams is central to geometric thinking.
While students certainly worked on questions that included diagrams, they rarely had to
produce one themselves. They were not required to describe triangles or ratios, and thus, the
language was only necessary because they needed to know which number to plug into the
formula.
Big Idea 2: Geometry is about working with variance and invariance, despite
appearing to be about theorems.
What I missed in my lessons is the opportunity to see the degree of invariance involved in right
angle trigonometry. When a triangle gets bigger, the sides change. When the angle changes, the
sides change. There is a lot of change occurring in the triangle. What is special are the things
that do not change, the invariance. However, instead of allowing my students to experience
variance and search for invariance, I told them what the invariance was. It wasn’t even special
to them. This is the danger of a lecture-based approach to the curriculum, students do not
experience the treasure found in invariance. To the student everything is invariance. Students
learn the invariance, not by discovery but by memorization. It is not surprising that students
cannot remember a simple thing like factoring or area of a triangle; they never construct it
and therefore rebuilding those ideas becomes impossible. The best they can hope for is a
mnemonic device to trigger their memory (examples, “FOIL”, “SOHCAHTOA”, “adds to b,
multiplies to c”).
Big Idea 3: Working with and on definitions is central to geometry.
Students in the lessons above were never required to work with a definition of sine. They
utilized sine ratio, but they didn’t work with or on it. Compare it to being given a hammer, told
that it is for driving nails and then asked to drive nails for a few hours. Chances are that you
would get really good at driving nails but you miss out on all the other things that hammer
could do. This is how students use trigonometric ratios. Show them how it works and then let
them do as the teacher does for a few hours. For years I started with a definition, often from
a textbook, and left no room for students to contribute to the building of the definition. We
were working from a definition, not towards it.
Big Idea 4: A written proof is the endpoint of the process of proving.
This is perhaps a missed opportunity teachers often make in the classroom. You would never
exempt an English student from writing a conclusion to their essay or a chemistry student
from writing their experimental conclusion. However, it is rare that a math student will get the
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Sinclair, N., & Pimm,
D. (2012). Developing
essential understanding
of geometry for teaching
mathematics in grades
9-12. Reston, VA: National
Council of Teachers of
Mathematics.
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opportunity to search, discover, conjecture, test and then formalize/generalize their conjecture
as a written proof. This rarely occurred in my classroom let alone in my trigonometry lessons.
Students believed that the ratios worked for all triangles because I told them that they work
in all triangles, end of story, no evidence needed.
So, after years of teaching trigonometry I realized that I was not effectively getting students
to grasp the concept of ratio. Despite repetitive practice with algorithms they seem to not
understand what they were calculating. I had a sense that if any of my unit plans needed to
change it would be this one. Thus, I tossed away my lessons and started from scratch.
One new approach I had to teaching trigonometry was piggy-backing it off of a topic that I had
felt was rather concrete. The month previous we had dedicated our time to linear functions.
More specifically we looked at the arithmetic nature of linear functions and the meaning
of slope. We spent days shifting, describing, drawing, and calculating slope. My students
experienced slope. If I were to give them a particular ratio, most could picture it in their mind
and sketch it out with relative precision. I felt this was an opportune time to begin asking
them if they could do the same with angle. And so, I gave them an angle and asked them to
visualize it and then draw it as accurately as possible. I realized that they probably had an even
greater ability to visualize angles than they did slopes and thus, the first trigonometric ratio,
tangent, had already been introduced and understood. I still wanted them to play around with
the relationship between angle and slope ratio so I introduced it by diagram. My initial plan
was to have them create the slope and angle but without Geometer’s Sketchpad I realized this
would be impossible. My options were to have them graph it on paper or have them play with
it on Sketchpad Explorer. The decision was really between the benefits of construction versus
the benefits of dynamic geometry and variance. Emphasizing Big Idea 2, I chose the latter and
constructed this sketch for them to use in Sketchpad Explorer.
There were three things I hoped to accomplish by introducing students to this exploration.
First, I hoped that students would recognize that steepness could be described in two ways. This
related to Big Idea 3 and more specifically addresses Essential Understanding 3a, “Geometric
objects can have different definitions. Some are better than others, and their worth depends
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both on context and values”. Here steepness could be described in two ways: slope described
the steepness relative to the horizontal axis and angle described it relative to any other line.
Both valuable concepts and both worthy of discussion in various contexts. Second, I hoped
to show them that we were working with triangles all along. Rise and run create the legs of a
right triangle, and it does not matter where you place those legs or how big they are, the slope
remains constant. In other words, I wanted them to see the invariance amongst the variance.
Third, I wanted to develop the invariance between angle and slope; no matter where you placed
B, slope was unique to that angle. This invariance allows us to use slope as a tool to derive angle
and make predictions, as was the case in question 7. Students began to make connections that
I had hoped. I overheard students use phrases like, “so 35.15°= 0.70”. I knew once that other
ratios were introduced students would need to correct this logic and notation.
The second part of my lesson was targeted to address Big Idea 2 and 3 and to necessitate a
change of language. Below is the sketch that students worked through in small groups.
The conversation that ensued was encouraging. While there were students who needed a lot
of direction on where to look, many dismissed the “slope” measurement that GSP calculated
and instead focused on “RISE/RUN”, noting that it stayed the same until the angle changed.
Students also began to argue over the term “rise” and “run”, suggesting that it was no longer
a “rise” or a “run”. While there was no consensus on what the names should be, one student
did offer the terms “opposite” and “beside”. This was a good start and I did not address it until
I felt the correct language was necessary.
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This, of course, gives rise to the third side of the triangle that some students had already begun
to address. While there were a few students who already knew the name of the third side,
none of them had thought about using it in a ratio like we did with opposite and adjacent.
For that purpose I created this lesson.
Once again I wanted to draw students’ attention to the value of the ratio and how that could
help determine the approximate angle. Questions 4 and 5 readdress the predictable nature of
the ratio and also helps students narrow their scope to look at one side at time.
If the instructions were followed correctly students should be able to determine that when you
keep adjacent constant and increase opposite and hypotenuse, the angle increases. Perhaps
more surprising for students is that when you keep opposite constant and increase adjacent
and hypotenuse, the angle actually decreases. This is something students rarely have the
opportunity to see, which is why most students couldn’t tell you why the cosine ratio increases
as the angle decreases. This sketch was followed by more instructions that would explicitly
have them determine the ratios.
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The use of Geometer’s Sketchpad throughout these activities gave students access to the
nature of the ratios and allowed them to spend more time to get a feel for how the ratios are
associated with the angles. These tasks were followed by a bit more terminology as students
felt it necessary to give names to these ratios so that they could remember what each one
described. However, before I shared “cosine” and “sine” I had students explain what the ratios
told them about the triangle. Students commented on the idea that any ratio could be used if
you knew two of the sides and employed the Pythagoras theorem. In a way, they suggested, all
of the ratios are part of a family. They also commented that the opposite/hypotenuse ratio went
from 0 to 1 and depending on the value you could estimate the angle. Meanwhile, adjacent/
hypotenuse ranged from 1 to 0 and could also be used to determine the angle. I suggested
that perhaps it would be easier to use a table that told us all these values instead of trying to
memorize them. And that is precisely what we did. For three days students used nothing but
trigonometry ratio tables to determine angle, and a calculator was later introduced as a device
that could store these tables for us.
As we came to the end of our unit, I have seen some students slip into the routine of
“SOHCAHTOA”, but there has been a number of difference in the students’ approach to
solving trigonometric problems. One difference I observed was that students understood the
range of values for each of the ratios. They understand why cosine and sine have a maximum
but tangent does not. The second difference was that students began looking for patterns. They
manipulate the triangles and look for clues that will help them determine future scenarios
(I even saw a student start his own table of ratios). Thirdly, trigonometry became more
geometrical and less algebraic. I was not interested in whether students could solve a ‘fill-inthe-blank’ algorithm. My questions were geared to estimation and patterning. Finally, students
seem to enjoy a personalized math experience. Trigonometry became something different than
the rest of the curriculum. They controlled the triangle, they controlled the values, and they
controlled the experience. While there were adaptations I would make, this new approach to
trigonometry contributes great value to not only the students’ understanding of trigonometric
ratios but more importantly an understanding about the nature of mathematics.
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