# A RESTRUCTURED TRIGONOMETRY COURSE

**1 Introduction**

College students take Trigonometry for a variety of reasons. Those concentrating
in

the sciences and engineering typically enroll in such a course to prepare
themselves

for Calculus and Differential Equations. Other students take Trigonometry to
sat-

isfy a university requirement that they show proficiency in College Algebra and
one

additional mathematics course. Many universities only offer one version of
Trigonom-

etry, so students find themselves in a course not necessarily designed for their
specific

needs. In this paper I propose an arrangement of topics which might better
engage a

diverse audience of students in a university trigonometry class.

**2 Demarcation of Trigonometric Topics**

College Trigonometry as we teach it today is primarily comprised of two broad
areas.

One deals with trigonometric functions on angles of triangles and the other with

trigonometric functions on the real line. In addition, trigonometry courses
typically

include some related topics such as polar coordinates, parametric curves,
analytic

geometry, or operations on complex numbers. The author is of the opinion that,

when arranging these topics for presentation, the two broad areas of
trigonometry

can and should be separated to reduce confusion.

To capture the students’ attention it makes sense to begin the course with trian-

gle trigonometry using degree measures of angles. College students have a
working

knowledge of the degree system even before they study trigonometry formally.
They

know what is meant by making a 90 degree turn and typically understand that
spin-

ning around once involves traversing 360 degrees. They are thus more or less
ready to

study trigonometric ratios of sides of a triangle whose angles are measured in
degrees.

There is really no need to complicate things by mentioning radians or
conversions

between degrees and radians in this first part of the trigonometry course.
Triangle

trigonometry can and should be done in its entirety before explaining to
students

that one radian has been traversed when a central angle of a circle intercepts
an arc

the same length as the radius of the circle. This non-intuitive definition
requires the

presentation of several new concepts that are not pertinent to the study of
triangles.

**3 Triangles**

When trigonometric functions are introduced as ratios of lengths of sides of a
right

triangle, a world of interesting applications present themselves. It is
relatively easy

for instructors to engage students as these relations are used to solve
accesible tri-

angle problems. Many text books currently in use begin with a presentation of
the

trigonometric functions on angles and then abruptly change their focus to
conversions

of units for measuring angles, graphs of trigonometric functions on the real
line, or

trigonometric identities [1], [2], and [3]. My experience is that some students
are left

baffled or overwhelmed by such an approach. To me it makes sense to proceed di-

rectly from right triangle trigonometry to the trigonometry of general
triangles. The

Laws of Sines and Cosines as well as Heron’s Formula can be presented, and many

problems of practical interest can be solved. It is also appropriate to
introduce the

trigonometric decomposition of planar vectors at this point in the course.
Interesting

results involving resultant forces and velocities can then be studied. The
practical

nature of this theory has more potential to engage the typical student than
other

topics in trigonometry that cannot be visualized to such a degree.

The presentation of trigonometric ratios in triangles can be made somewhat more

agreeable by downplaying cotangents, secants, and cosecants. These superfluous
func-

tions play an all too important role in many trigonometry text books. Though it
is

prudent to define them, it is counter productive to litter pages of text with
arcane

identities involving these trigonometric functions. Fortunately, manufacturers
of hand

held calculators are constrained by the number of buttons on their devices and
have

chosen to limit themselves to sines, cosines, and tangents. An argument could be

made for not presenting tangents either, but they are somewhat useful for
avoiding

mention of the hypotenuse in equations encountered in right triangle
trigonometry.

In addition, it is somewhat reassuring to have notation - in the form of an
inverse

tangent - for the distribution function of a Cauchy random variable [4, pp.
184-185].

As trigonometric functions are introduced, it is good practice to consider
30-60-90 and

isosceles right triangles in particular so that students have the opportunity to
evaluate

these functions on at least some angles with pencil and paper. As the instructor
shifts

to more practical problems, students must learn how to use trigonometric tables,

trigonometric functions on calculators, or trigonometric capabilities in a
computer

algebra system. In the discussion of trigonometric relations in right triangles,
only

acute angles are considered. Consequently, right triangle trigonometry should be

followed with a section on trigonometric functions evaluated on angles of more
than

90 degrees. The best approach is to define the functions on a domain of angles
in

standard position in the coordinate plane. Appropriate practice problems will
allow

the student to become proficient in evaluating the sine, cosine, and tangent of
angles

of all degree measures in the first revolution. There is really no need to treat
negative

angles or angles of more than 360 degrees here. Periodicity is best discussed in
the

second part of the course.

After students learn to evaluate the trigonometric functions on obtuse angles,
the

Laws of Sines and Cosines and their applications can be studied. Some texts
present

these trigonometric formulas for general triangles and then dwell on “solving”
trian-

gles and dealing with the ambiguous side-side-angle case. It could be argued
that

a trigonometry text should avoid this approach. When architects and engineers
use

trigonometry, they deal with tangible triangles. They measure certain angles or
sides

and determine the measures of other angles and sides applying trigonometric
laws.

Deducing whether a posited triangle with sides and angles of certain measures
exists

or is unique is a delightful puzzle to solve for some (and should be treated
briefly),

but many students are put off by such pursuits. When students have to worry as

to whether a triangle problem is even solvable, it detracts from the fun they
might

otherwise have deducing the width of a river or the pitches of the various
sections of

a busy roof.

**4. Trigonometric Functions on the Real Line
**

When the presentation of triangle trigonometry is complete, it is a good idea to start

right in with the trigonometric functions on the real line. One can begin with central

angles of circles and subtended arcs to introduce the concept of radians. After the

basic trigonometric functions are defined, the instructor can present the details of the

graphs of y = sin x, y = cos x, y = tan x, and y = arctan x. This is an excellent time

in the course to present the periodicity identities. After shifts, compressions, and

dilations have been presented, one can introduce linear combinations of trigonomet-

ric functions that arise as solutions of differential equations. Trigonometry students

typically are not familiar with derivatives, but their instructor can discuss how cer-

tain naturally occuring rates of change can be modeled by equations whose solutions

intuitively are periodic and smooth. The obvious illustrative example is a spring mass

system [5, pp. 217-219]. To derive

a sin(cx) + b cos(cx) = d sin(cx + h)

the identity for the sine of a sum is needed. To obtain this identity one can start with

the traditional derivation of the expansion of cos( α−β ), where α and β are central

angles of a unit circle. Then the the identities involving complementary angles and

the fact that the sine is odd and the cosine is even can be derived and used to obtain

sin( α + β ) and cos( α + β ). If these identities are introduced at this point in the course,

perhaps the student will appreciate their usefulness. It is only by rewriting the linear

combination of a sine and a cosine as a single sine function that one is able to deduce

the amplitute and phase shift of the modeled wave

The reduction, double, and half argument identites for the sine and cosine can follow

here so that those taking the course in preparation for Calculus will have this impor-

tant theory at their disposal. In addition, it is good practice to discuss the product

identities that prove useful when working with Fourier series [6, p. 294]. Many texts

over emphasize identities involving the other four trigonometric functions and tech-

niques for solving quadratic equations in all six of the trigonometric functions. This

educator feels that this is a poor use of time. In practice, the types of trigonometric

equations that one might need to solve are of the form cos(ax) = b or sin(ax) = b.

For example, a vector analysis of the the angle of inclination α that will allow an ath-

lete to throw a baseball a maximum horizontal distance, involves solving the equation

sin(2α ) = 1 [7, p. 873].

5. Supplementary Topics

5. Supplementary Topics

A solid introductory trigonometry class might include in its latter part the polar

coordinate sytem. In addition, students might be introduced to parametric equations

involving trigonometric functions and their associated planar curves. Knowledge of

these topics is essential for students who will eventually study Multidimensional and

Vector Calculus. If the instructor deems it important, time could also be spent at

the end of the course on rotations of conic sections. This topic is typically treated in

Linear Algebra, but could also be presented in a trigonometry course without using

matrix notation [8, p. 233]. Finally, the course could be concluded with DeMoivre’s

Theorem and its applications in the algebra of complex numbers.

**6. Concluding Remarks**

The author is of the opinion that trigonometry instructors following certain guide-

lines on the presentation of topics stand an improved chance of captivating a diverse

audience of students. At the same time, these guidelines will allow them to prepare

their students for further mathematical studies. Instructors should begin the course

with degree measures of angles and then treat triangle trigonometry in its entirety.

Only then should they discuss radians and trigonometric functions on the real line.

In addition, superfluous treatment of cotangents, secants,
and cosecants, should be

avoided. The essential identities should be treated at a point in the course
where

their usefulness is apparent, and less time should be spent on arcane identities
and

solving equations that are quadratic in trigonometric expressions. An outline of
the

proposed course is as follows:

**Triangle Trigonometry**

• Angles and Degree Measure

• Right Triangles and Definitions of Trigonometric Functins on Acute Angles

• Evaluating Trigonometric Functions on Special Angles

• Evaluating Trigonometric Functions on Angles in the First Revolution

• General Triangles and the Laws of Sines and Cosines

• Heron’s Formula and other Applications

• Trigonometric Decomposition of Planar Vectors and Resultants of Vector Sums

**Trigonometric Functions on the Real Line**

• The Definition of a Radian

• The Unit Circle and Graphs of the Sine, Cosine, Tangent, and Arctangent

• Shifts, Contractions, and Dilations of the Trigonometric Graphs

• Linear Combinations of Sines and Cosines as a Sine

• Derivation of the Sine and Cosine of a Sum of Arguments

• Essential Trigonometric Identities Involving Sines and Cosines

**Supplementary Trigonometric Topics**

• Polar Coordinates

• Parametric Curves Involving Trigonometric Expressions

• Rotations in Analytic Geometry

• DeMoivre’s Theorem and the Algebra of Complex Numbers

**To be Downplayed**

• Cotangents, Secants, and Cosecants

• Arcane Identities

• Solving Quadratic Equations in Trigonometric Expresions

References

[1] R. Larson and R. Hostetler, Trigonometry, Seventh Edition, Houghton-Mifflin,

2007.

[2] M. Sullivan, Algebra and Trigonometry, Eighth Edition, Prentice Hall, 2008.

[3] J. Coburn, Trigonometry, McGraw-Hill, 2008.

[4] M. DeGroot and M. Schervish, Probability and Statistics, Third Edition,
Addison-

Wesley, 2002.

[5] D. Zill and M. Cullen, Differential Equations With Boundary Value Problems,

Fifth Edition, Brooks/Cole, 2001.

[6] R. Johnsonbaugh and W. Pfaffenberger, Foundations of Mathematical Analysis,

Dover Publications, 2002.

[7] J. Stewart, Calculus Early Transcendentals, Fifth Edition, Brooks/Cole,
2003.

[8] L. Mirsky, An Introduction to Linear Algebra, Dover Publications, 1990.