This is the first in a series of articles about ways that one could teach trigonometry. We will not assume a specific setting in which the teaching will take place. So what we will say may be applicable to classroom settings or other settings. Also, we will not necessarily assume that the teacher is distinct from the student. As far as we are concerned, the teacher can also be the student. If you are teaching yourself trigonometry, then it is not too far-fetched to say that you are the teacher and the student.

So, what could teaching trigonometry involve? Is it about teaching the unit circle? Is it about making your students excited about trigonometric identities? Here’s what we could say: it depends on your students. The way you choose to present the subject depends on the level of preparation of your students. Have they had trigonometry before? If yes, then it might be fine to start with the technicalities right away. That’s probably the easy case. If no, then you might need to work on the settings. Devising a setting can be very helpful for the learners because it will help them put the subject in some context. Imagine you’re a student, you’ve never had trigonometry, and someone starts by explaining why $cos^2 x + sin^2 x = 1$. All this talk might be all gibberish to you, and this unfortunately might set the tone for the entire learning experience.

Devising a setting can be fun, so why not spend some time working on it. After all, your students may like it. Again, your audience will influence how you do it. Starting with some history of trigonometry can be entertaining. You don’t have to be a historian to briefly talk about the history of trigonometry. For this purpose, the web can be very helpful. If you don’t trust Wikipedia articles, then references at the end of Wikipedia articles can be a good start. If you really want to play safe, you can visit databases, like JSTOR or Google Scholar, which contain many articles published in journals. Again your audience matters. The audience for published articles in a journal may be different from your students, so you might need to skim these articles to see what information might be relevant. If you judge that your students may not understand the information in these articles, then preparing a nice presentation on the history of trigonometry might be a good idea. Remember many of your students may never have had trigonometry, so presenting the information clear of much technicality may be essential.

Focusing only on history may give your students the impression that trigonometry is just an old subject not worth their time and attention. At this point, those who may want to be engineers, scientists, and mathematicians may already be on the wrong track. To avoid this misconception, you may need to mention, with some level of generality, the use of trigonometry in some current examples. Construction of buildings and bridges involves trigonometric calculations, launching rockets into space involves trigonometric calculations, and sound waves can be represented as trigonometric functions. I bet that many of your students will be familiar with such examples; if not, you can find other ones. The web is a good place to run a search.

Remember that trigonometry may be entirely new to some of your students, so showing them a lot images and speaking in concrete terms may be helpful. Again, you put all of this effort to hopefully make them excited about the subject. For some of your students, this nice beginning may be a good encouragement to dig into some of the technicalities that will show up while practicing. If you teach at a school, then this introduction may motivate them to study and ultimately pass their exams.

Nowadays, talking about mathematics and ignoring technology may make one look uninformed. If you’ve been teaching mathematics, you’ve probably heard of the acronym STEM; the T is for “technology”. So mixing some technology with trigonometry even in this preliminary introduction to trigonometry wouldn’t hurt. After all, many of your students are probably computer whizzes. They’ve probably learned how to type on a computer before learning how to write with a pencil, and their second home may be the web. There are many resources on the web that combine trigonometry and technology. A nice one is GeoGebra. If you enter “trigonometry” in the search bar on site’s homepage, a long list of materials will appear.

Also, the way you start introducing the subject may influence the way your students perceive trigonometry. If you start on a stern note, they may think trigonometry will be very hard, so anxiety may paralyze some of them and may jeopardize their learning experience. But if you start cheerfully, then they may embrace the subject with that same cheerfulness, which may lead to a good experience for them. We would leave it to you to decide what’s appropriate. You can always make changes along the way.

Concerning time, you probably need to be as brief as possible for this introduction. What you could hope to accomplish is that your students, particularly those that are new to trigonometry, have a feel for the subject and somewhat get excited about it. Some of your students may spend more time on their own researching the subject. And since you will cover many topics in the future, you can have more opportunity to present other brief introductions on the topics at hand. If you teach at a school, probably one class or the first week of the term would be sufficient.

If your students have had trigonometry, then you probably won’t need to say much about history or introduce motivational examples because they may already know enough about trigonometry. But still, a very brief introduction can never hurt. At least, it could be a nice digression.

One last thing. Although your students will have much responsibility to their success in understanding trigonometry, you can also make the experience much fulfilling by giving them materials that may be useful to them. Be ready to make changes, as you may face scenarios that you will never have anticipated. Also be ready to have fun in this learning adventure.

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