How To Teach with DWFK Precalculus

Chapter 5: Analytic Trigonometry

This is a very important chapter for those who wish to pursue college mathematics, but not just because of the trigonometric material. The importance of equivalent expressions, the emphasis on proof, and the positive reinforcement gained from the ability to switch representations — all so important in analytic trigonometry — come as close to anything in the high school curriculum to approximating what matters to the real practitioners of mathematics. For this reason, we place deliberate emphasis on trigonometric identities as mathematical proofs and motivate them as such in the text.

Also, because their proofs are as important as their applications, we have placed the Law of Sines and the Law of Cosines in this section, along with the triangle area formulas.

Section 5.1 Section 5.4
Section 5.2 Section 5.5
Section 5.3 Section 5.6

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Section 5.1 Fundamental Identities

Objectives
You will be able to justify the fundamental identities and use them to simplify trigonometric expressions. You will be able to use them to solve certain trigonometric equations.

Key Ideas
Cofunction identities Odd-even identities
Domain of validity Pythagorean identities
Identity Trigonometric equation

Study Tips
The identities that follow directly from the triangle ratios or from the unit circle are called the Fundamental Identities. (Note that we define the concept of "identity" at the beginning of the chapter.) Eventually, you will need to memorize some identities whether they understand them or not; however, they should all be able to understand the Fundamental Identities.

Technology Tips
Graphing both sides of an identity to see if the graphs match can be a little tedious, but it is a nice way for you to verify their answers. Several other technology tips are shown in the section. Realistically, this section could be taught quite comfortably with no technology at all.
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Section 5.2 Proving Trigonometric Identities

Objectives
You will be able to decide whether an equation is an identity and will be able to prove identities analytically.

Key Ideas
Proof of an identity
Word ladder

Study Tips
Some people have lamented the decreased emphasis on proofs in the curriculum since the advent of certain education reforms, but it is only certain kinds of proofs that have been de-emphasized. Axiomatic algebra proofs (characteristic of the "New Math" of the 60's) have been de-emphasized because they did not contribute to student understanding, and two-column proofs in geometry were de-emphasized because they gave a misleading idea of what mathematical proofs look like. Really, trigonometric identities are the ideal introduction to proofs for beginners, as they actually read like proofs in higher mathematics courses. They have a structure that beginners can understand, and you can actually produce them on your own. This textbook embraces identity proofs as pedagogically important and teaches them as such.

For example, this kind of identity verification is never suggested:

Identity:    

"Proof":

Sometimes this sort of a thing is followed by a check sign, but it does not render any more educationally acceptable a mode of proof that begins with assuming what was to be proved and ends with a tautology. (We demonstrate a preferable alternative to this approach in Example 5.) Our emphasis is on constructing a logical path from what is known to what must be shown (as in the word ladders at the start of the section), as this is what mathematical proofs really look like.

Technology Tips
The main use of technology in this section is to provide graphical support for what is an identity and graphical refutation of what is not. You will not need to do this for all identities we ask them to prove. In particular, exercises 7–47 are true identities, and all that we require are the proofs.

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Section 5.3 Sum and Difference Identities

Objectives
You will understand the derivations of, and be able to apply, the formulas for the cosine, sine, and tangent of a difference or sum.

Key Ideas
Angle sum formula
Reduction formula

Study Tips
There is some debate about whether modern precalculus students need to memorize these (and other) formulas or not. The debate is not about whether students should memorize things, but rather about what they should be required to memorize and why. (These formulas can be stored on their calculators.) Regardless of where one stands on this debate, precalculus students ought to see how these formulas are derived. Moreover, calculus students will use these formulas later, and it saves time if one does not have to scroll down a calculator screen to find them.

Technology Tips
Most teachers realize that you can store these (and other) formulas on your graphing calculators, either as text screens or embedded in programs. At first glance you might think that this capability gives you an advantage on tests, but you need to consider that teachers are now less inclined to award ten points on an exam for simply stating a formula. A side effect of technology is that students and teachers alike are faced with finding more creative incentives for memorization.
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Section 5.4 Multiple-Angle Identities

Objectives
You will understand the derivations of, and be able to apply, the double-angle, half-angle, and power reducing identities.

Key Ideas
Double-angle identities
Half-angle identities
Power-reducing identities

Study Tips
The comments in the previous section about memorization apply equally well here. You will definitely use these formulas in calculus, especially the power-reducing formulas, which are the keys to finding certain antiderivatives.

Technology Tips
The trigonometric equations found in this section are intended as applications of the identities in the section, so solving them with calculators defeats their purpose. The instructions in the exercises should be carefully followed.
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Section 5.5 The Law of Sines

Objectives
You will be able to understand the proof of the Law of Sines and will be able to use the formula to solve a variety of problems.

Key Ideas
Law of Sines
Solving Triangles

Study Tips
The Law of Sines could equally well have appeared in the previous chapter, but we felt that it deserved to be among the identities. As with the other identities in this section, the derivation of the formula is a very good thing for precalculus students to see.

Technology Tips
Application problems of the kind featured in this section have always been part of a trigonometry course, but today's students can arrive at actual answers far more quickly (and more accurately!) than students of previous generations, thanks to technology. This is a good thing, as it allows you to concentrate on the formula rather than on the grubbiness of the computations. The main use of the calculator in this section is actually as a machine that calculates. As they say, go figure.
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Section 5.6 The Law of Cosines

Objectives
You will be able to understand the proof of the Law of Cosines and will be able to use the formula to solve a variety of problems. You will know Heron's formula and be able to use it to find areas of triangles and to solve appropriate application problems.

Key Ideas
Dihedral angle
Heron's formula
Law of Cosines

Study Tips
You ought to be able to follow the proof of the Law of Cosines, although the proof of Herons' formula might be a little heavy for some. (It's not that complicated, just a little intimidating with those five variables being pushed around.)

Technology Tips
If you are a good trigonometry student who likes to write calculator programs, you might enjoy the challenge of writing a calculator program that will solve triangles, using the Law of Sines and the Law of Cosines. The user would input three parts of a triangle (SSS or ASA or AAS) and the program would announce all six parts, having computed the missing ones by applying one of the laws. Let alone the difficulty of the programming, there are some subtle difficulties in the mathematics that make it hard to come up with a program that is correct in all cases. Good programmers should try to make the program "idiot-proof," i.e., designed to recognize and reject input values that will not determine a triangle (angles that sum to more than 180°, side lengths that do not satisfy the triangle inequality, etc.).