"For most of our students, trigonometric identities appear as if by magic." (Gordon, 2003, p.650) Isn't that the truth! I know that as a high school student I certainly thought that trigonometric identities simply appeared. I figured that someone somewhere understood why the identities were true, but I would never be able to understand it.
For the past five semesters, I have been working as a tutor in the math lab. I have found that one of the things that students have the hardest time with is remembering trigonometric identities. Almost half of the times that a Calculus II or Calculus III student has a question, it has to do with integrating a trigonometric function. Sometimes they remember that there is a trigonometric identity out there that will help them, but they don't remember it or remember where to find it. But, other times students don't think of identities at all. I feel like too often in trigonometry teachers present identities as a list of things to memorize for the chapter and then to forget later. Students who come in for trigonometry help typically don't understand the identities they are using, or why it applies to given situations. Students don't understand where the identities come from or how they relate to each other.
I remember being in that boat. I remember remembering that there was some identity somewhere that would apply, but I couldn't remember it. I remember one specific instance where I needed an identity that had a tangent squared in it. I knew that there was an identity that had a tangent squared and a one in it and some other trig function squared, but I had no idea what added to what. I could, however, remember that sin^2+cos^2=1. I had a teacher or classmate show me that if I simply divided the whole equation that I did know by cos^2, I would get the exact identity that I needed. This was a definite light bulb moment for me! I remember thinking, perhaps the other trig identities have a similar way to derive them. Now that I've immersed myself in the math world, I know that it is true! This is the purpose of my project. I wanted a way to show students and my classmates how the trigonometric identities relate to each other, and to show that we don't have to memorize them all!
This technology class has helped me to see that, as Charles Emenaker, author of "A Problem-Solving Based Mathematics Course and Elementary Teachers' Beliefs"(1996), said, "understanding concepts is more important than memorizing procedures" (p.78). The days of "But what if you don't have a calculator in your pocket...?" are long past! Everyone has a basic calculator in their pockets on their phones now, and most have access to the internet with just the touch of a button. Memorizing has less purpose when students have access so readily to all the information they want. Therefore, the important thing is to understand the formulas and know how to apply them to a given situation. "Mathematics teaching should emphasize activities that encourage students to explore mathematical topics; develop and refine their own ideas, strategies, and methods; and reflect on and discuss mathematical concepts and procedures. Searching for patterns, generalizations, connections, applications, verifications, and related problems must take precedence over rehearsing algorithms" (Garofalo, 1989,p.504).
This is why I chose to show some connections between the basic trigonometric identities that are often used in trigonometry, calculus and other classes. I present the sum and Pythagorean identities through geometric proofs, and then show how each of the others can be derived from these two basic sets of identities. I want students to have my light bulb moment. I want them to realize that the identities do come from somewhere, and that they all relate to each other. I want students to be confident deriving trigonometric identities on their own. Marsha Hurwitz, a high school math teacher, discussed this same topic in an article in "The Mathematics Teacher". She said, "Standard textbooks do cover the derivations of these expansions, but the results have a greater impact if students conjecture them first"(2002,p. 510) This is my goal. I want students to see how these basic identities can be derived, and then be able to apply this in the future when remembering identities or proving identities in trigonometry.