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# 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 , , and . 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

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

 R. Larson and R. Hostetler, Trigonometry, Seventh Edition, Houghton-Mifflin,
2007.

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

 J. Coburn, Trigonometry, McGraw-Hill, 2008.

 M. DeGroot and M. Schervish, Probability and Statistics, Third Edition, Addison-
Wesley, 2002.

 D. Zill and M. Cullen, Differential Equations With Boundary Value Problems,
Fifth Edition, Brooks/Cole, 2001.

 R. Johnsonbaugh and W. Pfaffenberger, Foundations of Mathematical Analysis,
Dover Publications, 2002.

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

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