How To Teach with DWFK Precalculus

Chapter 4: Trigonometric Functions

Three chapters of trigonometry have been condensed into two in this edition of the book, a move which we hope will allow more time for other topics which have increased in importance in recent years. (There has never been a problem adding such topics to textbooks; the problem has been that nothing has been subtracted!) The topics that remain have been retained for good reasons, and those reasons have motivated the way that the topics are presented. We will point those out in the Teaching Tips for each section.

Section 4.1 Section 4.4 Section 4.7
Section 4.2 Section 4.5 Section 4.8
Section 4.3 Section 4.6

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Section 4.1 Angles and Their Measures

Objectives
Students will be able to convert between radians and degrees, find arc lengths, convert to nautical miles, and solve problems involving angular motion.

Key Ideas
Angular motion Line of travel
Arc length of a circle Linear motion
Bearing Minute (angle measure)
Central angle Nautical mile
Course Radian
Degree Second (angle measure)

Teaching Tips
The difference between radians and degrees, and why it matters, are concepts not often understood by precalculus students, so we have attempted to motivate radians from the outset. It is very important that radians be used in calculus.

Note the "conversion factor" method used to solve the problem in Example 5. Students often overlook how they can make the units in an applied problem work for them.

Technology Tips
Be sure to have students do the angle conversions in the exercises on their own, as opposed to by simply pushing conversion buttons on their calculators. (They can use the calculators to do the computations.) This is a good time to introduce students to radian and degree "modes" on the calculator if they have not seen them before.

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Section 4.2 Trigonometric Functions of Acute Angles

Objectives
Students will be able to define the six trigonometric functions using the lengths of sides of a right triangle. They will be able to compute the six ratios without a calculator for isosceles right triangles and 30° – 60° – 90° triangles. They will be able to apply the ratios to solve problems that can be modeled geometrically with right triangles, including "solving" right triangles.

Key Ideas
30°60° – 90° triangle Secant (sec)
45° – 45° – 90° triangle Similar triangles
Cosecant (csc) Sine (sin)
Cosine (cos) Solving a triangle
Cotangent (cot) Standard position of an angle
Right triangle trigonometry Tangent (tan)

Teaching Tips
The triangle ratios are at the heart of any definition of trigonometric functions, so we begin with them. As the two "famous triangles" are still considered basic knowledge for students of trigonometry, we stipulate that students should be able to find those ratios without a calculator. There will be other occasions throughout the chapter that we will set calculators aside, and we will explain why in each case as it arises.

The modeling exercises in this section are just the beginning. There will be many more such problems throughout the two trigonometric chapters.

Technology Tips
This section is chock full with technology tips, gathered into the subsection "Evaluating Trigonometric Functions with a Calculator." We recommend that teachers and students pay close attention to the material in that short subsection.
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Section 4.3 Trigonometry Extended: The Circular Functions

Objectives
Students will understand how the trigonometric functions are extended by the unit circle to become functions of any angle. They will be able to use reference triangles to find trigonometric functions of real numbers.

Key Ideas
Measure of an angle Reference triangle
Initial side of an angle Quadrantal angle
Terminal side of an angle Unit circle
Positive and negative angles Periodic function
Standard position of an angle Period of a function
Coterminal angles Circular functions

Teaching Tips
Although calculators replaced trig tables quite a few years ago for the evaluation of trig functions, the concept of "reference angle" has strangely endured. (Finding the reference angle was traditionally the first step in preparing to use a trig table.) We have replaced this arguably outmoded concept with the "reference triangle," which links the circular functions more directly to the unit circle. The reference triangle is drawn in its appropriate quadrant, and all six ratios are read directly from the triangle with their signs attached.

The more students understand about the unit circle, the less they will need to depend on memorization to work with trigonometric functions. Exploration 2 is a good group activity to foster that understanding.

Technology Tips
This is a section in which the calculator is hardly a factor at all. In fact, students will wonder why they are bothering with unit circles and reference triangles to compute something like , a number that can be found easily on a calculator. The note at the bottom of page 360 attempts to answer this reasonable question.

Note, too, that the instructions for many of the exercises in this section specify that they are to be done without calculators.
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Section 4.4 Graphs of Sine and Cosine: Sinusoids

Objectives
Students will be able to generate the graphs of the sine and cosine functions and explore various transformations of these graphs, called sinusoids, algebraically and geometrically. They will be able to model periodic behavior with sinusoids and thereby solve real-world problems.

Key Ideas
Amplitude of a sinusoid Phase shift of a sinusoid
Period of a sinusoid Sinusoid

Teaching Tips
Some students will have seen modeling problems based on right triangle trigonometry in previous courses (and all should have seen them in Section 4.1), but few are likely to have used circular functions to model periodic behavior. The ability to generate sinusoids instantly on graphing calculators helps students to see the effects of the various transformations. Once they figure them out, they should be able to write equations of sinusoids with specified amplitudes, periods, and phase shifts before confirming their results on their calculators.

Technology Tips
An interesting way to introduce sinusoids is to ask students working in pairs to produce a graph that looks exactly like this in the window shown:

They will have to figure out by trial-and-error how to turn a sine function upside down, double its frequency, and triple its amplitude. (You could also do this exploration in steps, presenting them with the graphs that show one transformation at a time.)

Note that the instructions for many of the exercises in this section specify that they are to be done without calculators. This is important so that students will get practice in moving between the graphical and algebraic representations of sinusoids.
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Section 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

Objectives
Students will be able to generate the graphs of the tangent, cotangent, secant, and cosecant functions and explore various transformations of these graphs.

Key Ideas
Cosecant function Secant function
Cotangent function Tangent function

Teaching Tips
Although the tangent function is not one of the Ten Basic Functions from Chapter 1, it is an important function for students to know well. (Indeed, if we had opted for a dozen basic functions, tangent would have been #11. The greatest integer function would have been #12. We finally opted for a less intimidating ten.) You might want to see how many of the tangent function properties on page 379 your students can come up with on their own before looking at the list. They can construct their own lists of properties for cotangent, secant, and cosecant.

Technology Tips
Students will probably not have buttons on their calculators for the cotangent, secant, and cosecant functions. Make sure that they do not try to use the buttons for , a mistake about which they have been warned in several places. See the margin note on page 381.

Graphing y = tan x will invariably produce those "false asymptotes" on the calculator, since the asymptotes occur at irrational values. If you want to graph the tangent function without seeing the false asymptotes, you can fool the grapher with a clever horizontal shrink:

The same horizontal shrink in the same window eliminates the false asymptotes from the graphs of cotangent, secant, and cosecant (listed in the screen above as Y2, Y3, and Y4 respectively).
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Section 4.6 Graphs of Composite Trigonometric Functions

Objectives
Students will be able to generate the graphs of sums, differences, and other composite functions that involve trigonometric functions. They will be able to model damped oscillation with a composite trigonometric function.

Key Ideas
Damped oscillation
Damping factor

Teaching Tips
This section could easily have been much longer were it not for the fact that it is only necessary to give a brief glimpse at composite trigonometric functions at this level. Students can generate all kinds of bizarre waves on their calculators by combining trigonometric functions in various ways of their own choosing, a luxury not available to the previous generation. Not much is gained, however, in analyzing them with precalculus tools.

Technology Tips
A graphing calculator curiosity that has tantalized math teachers over the years is the unexpected way that periodicity affects graphs with high frequencies. For example, the TI-83 and TI-82 calculators, which have 95 columns of pixels across the screen horizontally, will produce the following graph of the function in the screen shown:

The graph of does not show up at all (since every x-value the calculator plugs in results in a y-value of 0). The key to the paradox is that the increment along the It's the same phenomenon that makes [-4.7, 4.7] yield a "decimal" window. Fiddling with this is fun. Figuring out what is really going on is a challenge.
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Section 4.7 Inverse Trigonometric Functions

Objectives
Students will be able to analyze the properties of the inverse trigonometric functions by deriving them from the corresponding properties of the trigonometric functions. Students will be able to produce the graphs of and simplify expressions involving compositions of trigonometric and inverse trigonometric functions.

Key Ideas
Inverse cosine function (arccosine)
Inverse sine function (arcsine)
Inverse tangent function (arctangent)

Teaching Tips
The inverse trigonometric functions that are needed for calculus are the three covered in this section — and they are needed. The connection to the original trig functions provides a nice, spiraled review of inverse functions in general, while the imposed domain restrictions serve as a nice reinforcement of "unit circle thinking."

Exploration 1 and Example 5 are particularly important in terms of calculus preparation. The most mysterious feature of the calculus of inverse trig functions is the way they appear out of nowhere when dealing with functions like , a feature that is explained by understanding how the functions behave, not by understanding the calculus.

Technology Tips
You can graph the "whole" inverse relation of a trigonometric function in parametric mode, then trim the curve down to the one-to-one section by restricting the parameter T. For example, here is how we can trim down the arcsine:

(The missing values for Ymax and Yscl on the middle screens are , respectively.)

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Section 4.8 Solving Problems with Trigonometry

Objectives
Students will be able to model real-world problems with trigonometric functions and thereby solve them.

Key Ideas
Angle of elevation
Angle of depression
Simple harmonic motion

Teaching Tips
There are not many new things to be learned in this section; it is mostly an opportunity for students to put together their knowledge of trigonometry thus far and use it to solve genuine problems. The problems either rely on the right triangle ratios or on modeling periodic behavior with sinusoids. Another round of applications will follow the various identities in the next chapter.

Technology Tips
In this section, it is "no holds barred" as far as technology is concerned. This book attempts to distinguish between exercises that are intended to teach skills and exercises that are intended to see if you can arrive at the correct answer. Modeling problems are almost always of the latter type. Students should be told that the object is to find the correct answer — just as in the real world — and that they should feel free to use all problem-solving tools at their disposal.

We should also tell students, just in case, that finding the answer in the back of the book is not the kind of problem-solving tool we have in mind. They should be encouraged to check their answers, but they need to be able to produce independently the solutions that lead to them.