|Section 4.1||Section 4.4||Section 4.7|
|Section 4.2||Section 4.5||Section 4.8|
|Section 4.3||Section 4.6|
|Angular motion||Line of travel|
|Arc length of a circle||Linear motion|
|Bearing||Minute (angle measure)|
|Central angle||Nautical mile|
|Degree||Second (angle measure)|
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.
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.
|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)|
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.
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.
|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|
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.
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.
|Amplitude of a sinusoid||Phase shift of a sinusoid|
|Period of a sinusoid||Sinusoid|
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.
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.
|Cosecant function||Secant function|
|Cotangent function||Tangent function|
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.
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).
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.
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:
Inverse cosine function (arccosine)
Inverse sine function (arcsine)
Inverse tangent function (arctangent)
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.
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.)
Angle of elevation
Angle of depression
Simple harmonic motion
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.
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.