10 Things I Wish I Would Have Learned In Trigonometry Class

Okay, so I'm not going to write a list, sorry to disappoint. However, during my MTH 229 review I realized that I somehow missed a lot of really cool foundational ideas from trigonometry. It is a subject that I have always found fascinating, but also confusing. I remember asking questions, like "where does the sine graph come from," and getting really terrible answers. I remember being really overwhelmed with all of the identities and rules, but I think that is because I never really developed an understanding of why things work the way they do; I was just taught to memorize a few things and learned how to manipulate them.  So what I want to do here is spend a little time thinking about the things I wish I had a better understanding of early on, and perhaps some good ways to teach those things.

As I said, I memorized a few things, basically enough to help me solve the problems I was given. As an example of this I'm going to share something a little scary and embarrassing; until about a week ago I couldn't have properly explained why a circle is represented by the equation x ² + y ² = r ². I could use the equation to draw a graph and vice versa, but lacked the true understanding of why a circle was defined as such. The concept isn't hard, but I don't remember being taught, and I guess I was never aware that my knowledge was not understanding. One thing I do remember is using these 'special triangles,' the 30-60-90, and the 45-45-90 triangles pictured. For some reason I always remember these ratios and could use related trig functions to solve problems as necessary. We can see that they are related to the unit circle, which I vaguely remember being taught, but the triangles seem so much easier. The unit circle is so intimidating, especially to someone new to trigonometry, just look at that thing with all those square roots...yuck. I feel like the information is accessible if you know how to use it, but having to reconstruct or memorize it seems like a useless pain. I guess I tended toward the triangles because the made more sense to me, were easy to construct, and contained much of the same information as the unit circle.

So what I want to share now are a few ideas and resources that have helped me develop a much more thorough understanding of trigonometry over the last couple weeks. I'll begin with an article that discusses a way to introduce the topic of trigonometry without mentioning trigonometric functions. The teacher here assigns their students the task of drawing angles (every 10 degrees) within a circle of radius one, measuring the x and y values where the angle intersects the circle, and calculating the ratio of x/y. Students are encouraged to use shortcuts when they find them available, but also have the option of drawing each angle and calculating each value. Though it may seem a little tedious, I think this exercise would give students an opportunity to develop real understanding of what the trig functions are, and how they may be used. From here you would obviously introduce sine and cosine functions as well, you could compare ratios as the angle changes, and make connections amongst the functions. I recently used trig tables for the first time to do some basic calculations. I found them to be easy to use, and just looking at the tables gives you the opportunity to see a lot of patterns related to the functions.




Another resource I really like includes the diagram to the right; this particular diagram shows many of the trig identities built around the circle of radius one. The article that goes along with this visual actually explains it all really well using an analogy of a dome, a wall, and the ceiling. It does this in a way that allows us to construct, as opposed to just memorize, the diagrams and functions. The whole idea here is to build the understanding by making connections to the way these functions work and interact with each other. For example, it goes on to explain that Sine and Cosine are just percentages of the hypotenuse, which I think is a great strategy. It also includes a link to an interactive graph with a unit circle, graphs of the six common trig functions, as well as numerical representations of each of the functions. It is a really cool tool for discovery and understanding.



Finally, I want to talk about this interactive unit circle for sine and cosine. When I first played with it I though, oh yeah, this is pretty cool. Then I went back to it again and tried to really take in all the information that was being displayed; I was amazed. There is something really awesome about these interactive tools that allow you to see way more information than just plotting points on a graph; the ability to simultaneously see the circle, graph, and triangle created through all possible ranges is unbelievably helpful in building understanding. This is the graphic, that now famously, gave me understanding of the equation of a circle; I was just trying to figure out everything that was going on and it hit me. The screenshot below shows just some of the information available, but gives you an idea of the deep connections that can be made from the material. I think students could definitely benefit by doing some investigative work with this and making connections between the circle and triangles built within; I think they could either be given questions to answer, or even come up with questions of their own based on correlations they see. If nothing else, this is a great way to develop understanding of the sine and cosine graphs (another thing I historically struggled with). Again, as someone who has worked with the information in this graph quite a bit, spending a little time exploring helped deepen my understanding and build new connections between representations. Of all the learning tools and techniques I have discussed thus far, this is one that I will definitely use in teaching trigonometry.