This is the first in a series of articles about ways that one could teach trigonometry:

On Teaching Trigonometry: Angles and the Right Triangle
On Teaching Trigonometry: Trigonometric Functions and the Unit Circle
On Teaching Trigonometry: Graphing Trigonometric Functions (Part 1)
On Teaching Trigonometry: Graphing Trigonometric Functions (Part 2)

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 ent...