2.
Vol. 102, No. 2 • September 2008 | Mathematics Teacher 145
found only if two sides of a right triangle having an
angle of 15° were given. A ratio understanding of
sine would also not permit students to determine
in which quadrants sine was increasing or to graph
sin 2x. Many calculus tasks, such as determining
the derivative of sin x, make little sense with a ratio
understanding of sine.
To address the latter tasks, students need to
have a function-based understanding of trigono-
metric operations (Weber 2005). They need to
understand operations such as sine as a process
that takes an angle as an input and maps this angle
to a real number. To understand a trigonometric
operation as a function, students need to know a
process they can use to evaluate that function for
any given angle, and they must be able to anticipate
the approximate result of that method and reason
about the properties of that result without actually
performing the steps of the process.
TRADITIONAL INSTRUCTION OF TRIGONO-
METRIC OPERATIONS
Too often trigonometry instruction emphasizes
procedural, paper-and-pencil skills at the expense of
deep understanding (Hirsch, Weinhold, and Nichols
1991). An inspection of several popular high school
algebra, geometry, and trigonometry textbooks (e.g.,
Hollowell, Schultz, and Ellis 1997; Schoen 1990;
Larson 2004) reveals that students are typically
first taught trigonometric operations as ratios (e.g.,
sin q is defined as y/r in a labeled right triangle or
as “opposite over hypotenuse”). Students are asked
to use these ratios to accomplish tasks such as those
suggested in figures 1 and 2 and then solve word
problems. After several sections devoted to these
topics, the textbooks introduce the unit circle model
of trigonometric functions. At this point, the texts
sometimes ask students to imagine applying a pro-
cess to find sine and cosine of a particular angle
(such as walking r units around the unit circle and
locating the y- and x-values, respectively, of the stop-
ping point). However, students are not given the
opportunity to apply this process. The exercises they
are asked to complete rarely require a process under-
standing of trigonometric operations; most simply
require using a ratio conception of the trigonometric
operations or applying algebraic techniques. Other
researchers note that traditional instruction empha-
sizes understanding trigonometric functions as ratios
and does not enable students to understand them as
functions (Kendal and Stacey 1997).
THE EFFECT OF TRADITIONAL INSTRUCTION
ON STUDENTS’ UNDERSTANDING OF
TRIGONOMETRIC OPERATIONS
To investigate students’ understanding of trigono-
metric operations, I conducted a study of thirty-one
students who were completing a college trigonom-
etry class. I asked them to take a test without using
a calculator and then invited four students for
an interview. The teacher of the class, who was
unaffiliated with my study, described his teaching
as traditional; in particular, he claimed to teach
primarily by lectures based largely on the course
textbook, with an emphasis on students develop-
ing procedural skills. The textbook for the course
was Lial, Hornsby, and Schneider (2001), which
was structured similarly to the textbooks described
earlier. The full results and methodology of this
study are reported in Weber (2005). The purpose
of reporting these results here is to illustrate the
limited understanding of trigonometric functions
many students have after completing a trigonom-
etry course.
Two of the test questions were these:
1. Approximate cos 340°.
2. For what values of x is sin x decreasing? Why?
Only six of the thirty-one students estimated
that cos 340° was between 0.5 and 1.0. Nine stu-
dents correctly noted that sin x was decreasing for
90° < x < 270°, and six offered a convincing expla-
nation for why this was the case. These results sug-
gest that these students had difficulty understand-
ing trigonometric functions.
Fig. 1 A labeled right triangle
Fig. 2 A right triangle with a given angle, a given side, and
two missing sides
5 3
A
4
10 a
30°
b
40°
40°
70° 160°
The intersection
point is about
0.75 to the right
of the y-axis.
85° 230°
(0, 1)
4
10 a
30°
b
40°
40°
The intersection
point is about
0.75 to the right
of the y-axis.
The intersection
point is about
0.65 above the
x-axis.
3.
146 Mathematics Teacher | Vol. 102, No. 2 • September 2008
My interviews with a subset of the students
revealed some reasons why. Consider the tran-
scripts below (all students’ names are pseudonyms):
Interviewer: Describe sin x for me in your own
words.
Steve: To find sine, it would depend on the problem
that was given to me. If I was given a triangle,
I would divide y and r. If I were given one of
the special angles, like 30°, 45°, or 60°, I would
have this number memorized. There are other
problems which can be solved by reference
angles, or using formulas, like sin q + cos q = 1
[sic]. How you find the answer depends on how
the problem is worded.
Interviewer: What can you tell me about sin 170°? Can
you give me an approximation for this number?
Steve: I don’t ... I would need the triangle. Maybe if you
told me what some other value of sine would be, or,
like, what cos 170° would be, I could find what sine
is. Otherwise, I would need to use a calculator.
Steve’s responses were representative of those
of the four students I interviewed. Steve seemed
to view sine and cosine as algorithms (based on
ratios or on algebraic manipulations) that could
be applied but only if other information, such as a
labeled right triangle, was also provided. Without
this information, Steve could not conceive of how
to apply the sine or cosine operation to an angle,
except for special cases. (Because my tested sample
consisted of only one class of students, we must not
generalize these results inappropriately.)
AN ALTERNATIVE APPROACH TO
TEACHING TRIGONOMETRY
The following suggested instruction is based on
the idea that trigonometric operations such as sine
can be understood as geometric processes. One pro-
cess for computing sine is to construct a unit circle
on a Cartesian plane, use a protractor to draw a
ray emanating from the origin such that the angle
between the positive half of the x-axis and the
ray is the input angle, locate the point of intersec-
tion between the ray and the circle, and determine
the y-value, or height, of that intersection. Recent
research in mathematics education indicates that
students have difficulty imagining the application of
a process without the experience of actually applying
it. Instead, students may best develop a deeper
understanding of processes by first applying them
and then reflecting on their actions (Tall et al. 2000).
Here I describe instruction that I designed and
then implemented in a college trigonometry class
that I taught. (One typical lesson is provided in the
appendix.) To involve students kinesthetically in
computing sines and cosines, each student was given
a protractor and a unit circle drawn on graph paper
and marked with Cartesian coordinates such that ten
tick marks constituted one unit. I then described and
modeled a procedure for computing sines and cosines
by using a protractor to draw an angle with vertex
at the origin and one ray along the x-axis, marking
where the other ray intersected the circle, and using
the tick marks to estimate the x- or y-coordinate of
that point. This procedure was also fully described
in a student handout. Students worked in groups on
a series of classroom activities in which they found
the sines and cosines of six angles by using the proce-
dure. While they worked, I circulated among them to
answer any questions they might have and make sure
they were applying the procedure correctly.
Students were next asked to evaluate the sines
and cosines of some angles by anticipating the
results of the procedure without applying the
procedure itself. For instance, students could find
sin 270° by realizing that they needed to examine
where the bottom half of the y-axis intersected
the unit circle. Students were also asked to make
judgments about the procedure’s output without
actually applying it. For instance, they were asked
to determine which was greater—sin 23° or sin 37°.
These tasks helped students reason about the
process in general and make judgments about sines
and cosines without applying each step.
I used lessons of this type throughout the course.
For instance, students learned how to find sines,
cosines, and tangents by constructing right trian-
gles on a Cartesian plane, measuring the lengths of
the sides, and computing ratios. Once the students
understood this procedure, they could perform
tasks like those required in figure 1, because they
understood the essentials; it was as if someone had
created a triangle and measured the lengths of the
sides for them. However, students’ understanding
of the functions was not restricted to reasoning
from diagrams; they could imagine producing the
diagrams themselves, if necessary. When studying
reference angles, students were first asked to draw
the desired angle, find the appropriate reference
angle themselves, and then compute the sines and
cosines by looking at it. Students instructed in this
geometric approach understood reference angles
easily and did not need to rely on the mnemonic
strategies that trigonometry students commonly
use. A more thorough description of this instruc-
tion is provided in Weber (2005).
Toward the end of the course, I gave the forty
students in my class the same paper-and-pencil
test I had given to the students in the lecture-based
class described earlier. When asked to approximate
cos 340°, thirty-seven of the forty students gave a
response of some number between 0.5 and 1. When
asked for those values for which sin x was decreas-
4.
Vol. 102, No. 2 • September 2008 | Mathematics Teacher 147
ing, thirty-four students gave a correct answer, and
thirty-two provided an adequate justification.
On the basis of the students’ test responses, I
interviewed four students whose abilities varied
(one very good student, two average students, and
one struggling student) and whose responses were
typical of those of other students. Throughout
these interviews, the students were able to explain
properties of the sine function by reasoning about
the process of computing sines. Two representative
excerpts of these interviews follow:
Interviewer: Why is sin x a function?
John: Because for, uh, each angle, there’s … going
back to the unit circle, if you put something in
for the sin x, it’s only going to cross at one point.
Each angle is going to be one angle, and that
one angle is going to cross the unit circle at one
point. That one point is going to have a y-value.
It will have one and only one y-value.
Note that John refers to the process used to com-
pute sines to justify why sine has a certain prop-
erty. Three of the four students interviewed gave
similar responses. It is also worth noting that none
of the four interviewed students in the traditionally
taught class could justify why sin x was a function,
even after being told that an operation was a func-
tion if each input had a unique output.
In the following excerpt, Erica is able to use her
process conception of sine to approximate sin 170°:
Interviewer: What can you tell me about sin 170°? Can
you give me an approximation for this number?
Erica: The answer would, oh, be, I’d say, 0.1.
Interviewer: That is a good guess. How did you get
that answer?
Erica: I pictured making a 170° angle with
a protractor and seeing where the angle
intersected the circle.
Interviewer: I see. And how did you know it would
intersect at 0.1?
Erica [drawing a diagram]: Well, it would intersect
right there [pointing to the point of intersection].
Here Erica is able to show how she can adapt her
understanding of a process used to compute sines
to approximate sin x quickly and accurately for an
arbitrary angle x.
These results are from only one class, one that
I taught myself, so it is important not to generalize
these results inappropriately. These results,
however, do show that a geometric approach
to trigonometry can be effective in developing
students’ understanding of trigonometric
operations and suggest that this approach could be
applied in other classrooms.
DISCUSSION
The data from this research suggest that the geo-
metric approach to trigonometry can lead students
to understand trigonometric operations as func-
tions while traditional instruction does not. In this
section, I describe the features that distinguish this
approach from the lessons that appear in many
popular high school textbooks. The first is the
emphasis placed on performing a geometric process
to compute sines, cosines, and tangents physically. I
examined the way in which several high school and
college textbooks presented trigonometric functions
and found that most mentioned a process (using the
unit circle) only in passing. The questions students
were asked to complete could almost always be
accomplished by treating trigonometric operations
as ratios. Second, the alternative approach asks stu-
dents to perform this process physically and reflect
on their actions. The textbooks provide neither of
these two features.
It is tempting to say that the emphasis on the unit
circle was the reason students instructed in the alter-
native approach did better, but in a large-scale study,
Kendal and Stacey (1997) found otherwise. They
compared students’ learning in classes in which a
right-triangle model was used with students’ learn-
ing in classes in which a unit-circle model was used.
Students who were taught the right-triangle method
performed significantly better on a subsequent post-
test than students who were given the unit-circle
model. So simply teaching about trigonometric oper-
ations by using a unit circle model is no guarantee
that substantial learning will occur. However, giving
students the opportunity to think of sine and cosine
as processes is critical, regardless of the model used
to teach these operations.
If students are given the opportunity to apply
and reflect on the constructive geometric processes
used in evaluating trigonometric operations, they
will understand these operations at a much deeper
level than if they are taught that these operations
are merely ratios that can be applied to given right
triangles. One appealing aspect of the geometric
approach is that implementing it does not require
a teacher to alter his or her classroom radically.
No special technology or training in special teach-
ing methods is required. Implementing the ideas
expounded here offers practicing teachers the
opportunity to create an active, collaborative, hands-
on learning environment that has the potential to
help students understand trigonometric concepts.
REFERENCES
Blackett, N., and D. O. Tall. “Gender and the Versa-
tile Learning of Trigonometry Using Computer
Software.” In Proceedings of the 15th Meeting of
the International Group for the Psychology of Math-
5.
148 Mathematics Teacher | Vol. 102, No. 2 • September 2008
Tall, D. O., M. Thomas, G. Davis, E. Gray, and A.
Simpson. “What Is the Object of the Encapsulation
of a Process?” Journal of Mathematical Behavior 18,
no. 2 (2000): 1–19.
Weber, Keith. “Students’ Understanding of Trigo-
nometric Functions.” Mathematics Education
Research Journal 17, no. 3 (2005): 94–115. ∞
ematics Education 1, edited by F. Furinghetti, pp.
144–51. Assisi, Italy. 1991.
Hirsch, Christian R., Marcia Weinhold, and Cameron
Nichols. “Trigonometry Today.” Mathematics
Teacher 84, no. 2 (1991): 98–106.
Hollowell, K. A., J. E. Schultz, and W. Ellis Jr. HRW
Geometry. Austin, TX: Holt, Rinehart, and Win-
ston, 1997.
Kendal, M., and K. Stacey. “Teaching Trigonometry.”
Vinculum 34, no. 1 (1997): 4–8.
Larson, R. Trigonometry. Boston: Houghton Mifflin, 2004.
Lial, M. L., J., Hornsby, and D. I. Schneider. College
Algebra and Trigonometry. Menlo Park, CA:
Addison Wellesley, 2001.
National Council of Teachers of Mathematics
(NCTM). Principles and Standards for School
Mathematics. Reston, VA: NCTM, 2000.
Schoen, H. Trigonometry and Its Applications.
Columbus, OH: Glencoe/McGraw Hill, 1990.
KEITH WEBER, keith.weber@gse.
rutgers.edu, is an assistant professor
of mathematics education at Rutgers
University in New Brunswick, New
Jersey. He is interested in how high school and
college students learn mathematics, including
trigonometry and proof. Photograph by Erin Maguire;
all rights reserved
APPENDIX
Computing Sines and Cosines by Using the Unit Circle
1. Start with a unit circle drawn on a Cartesian graph. A unit circle is a circle with a radius of 1 whose
center is the origin.
5 3
A
4
10 a
30°
b
40°
(0, 1)
2. Use your protractor to make an angle with respect to the positive part of the x-axis. A 40° angle is
shown here.
5 3
A
4
10 a
30°
b
40°
70°
6.
Vol. 102, No. 2 • September 2008 | Mathematics Teacher 149
3. Locate the point of intersection between the ray you have just drawn to complete the angle and the
unit circle. Using a ruler or your graph paper, find the coordinates of your intersection. The x-value
is the cosine of the angle you have constructed, and the y-value is the sine. In this case, sin 40º is
about 0.65, and cos 40º is about 0.75.
40°
70° 1
The intersection
point is about
0.75 to the right
of the y-axis.
The intersection
point is about
0.65 above the
x-axis.
85°
Classroom Exercises
1. Compute the following sines and cosines by using a protractor and a unit circle:
(a) sin 30° and cos 30° (b) sin 170° and cos 170°
(c) sin 120° and cos 120° (d) sin 260° and cos 260°
(e) sin 80° and cos 80° (f) sin 325° and cos 325°
2. Without explicitly computing these values (i.e., without using a protractor and a ruler), compute
the following sines and cosines. To get you started, I will do the first exercise for you.
(a) sin 90° and cos 90° (see diagram below)
Explanation: A 90° angle is a right angle. I draw a 90° angle inside the unit circle. It intersects the
circle at the top of the circle. This point is (0, 1). So sin 90° = 1 and cos 90° = 0.
5 3
A
4
10 a
30°
b
40°
40°
70° 160°
The intersection
point is about
0.75 to the right
of the y-axis.
(0, 1)
(b) sin 0° and cos 0° (c) sin 180° and cos 180°
(d) sin 270° and cos 270° (e) sin 360° and cos 360°
7.
150 Mathematics Teacher | Vol. 102, No. 2 • September 2008
3. Approximate the sine and cosine of the angles drawn in the diagrams below:
40°
40°
70° 160°
The intersection
point is about
0.75 to the right
of the y-axis.
The intersection
point is about
0.65 above the
x-axis.
85° 230°
4. Without doing the computations, answer the following questions. Justify your answer.
(a) Is sin 140° a positive number or a negative number? (Hint: Draw a unit circle and approximate a
140° angle).
(b) Is cos 200° a positive number or a negative number?
(c) Which is bigger—sin 23° or sin 37°?
(d) Which is bigger—cos 300° or cos 330°?
Homework Questions
1. Compute the following sines and cosines by using a protractor and a unit circle:
(a) sin 50° and cos 50° (b) sin 127° and cos 127°
(c) sin 200° and cos 200° (d) sin 300° and cos 300°
2. Approximate the sine and cosine of the angles drawn in the diagrams below:
A
4
10 a
30°
b
40°
40°
70° 160°
The intersection
point is about
0.75 to the right
of the y-axis.
The intersection
point is about
0.65 above the
x-axis.
85° 230°
(0, 1)
3. Without doing the computations, answer the following questions. Justify your answer.
(a) Is sin 240° a positive number or a negative number? (Hint: Draw a unit circle and approximate a
240° angle.)
(b) Is cos 300° a positive number or a negative number?
(c) Which is bigger—sin130° or sin 147°?
(d) Which is bigger—cos30° or cos 230°?
4. In what quadrants will sin x be positive? In what quadrants will cos x be positive?
5. Can you find an angle x so that sin x = 2? If so, what angle is it? If you cannot find such an angle,
why not?
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