Round these internet parts, a lot of people are fans of Kate and Riley's intro to unit circle trigonometry. Their introduction is designed to get students seeing that any point on the edge of the unit circle is (cosA, sinA). Mine is too, but I go about is slightly differently, and it seems to work.
In short: My first goal when teaching trigonometry is to change the way that my students see circles. When they see a circle, I want them to see it as full of right triangles. Then we spend a day talking about special right triangles inside circles.
We start with drawing a lot of right triangles:
And eventually kids end up with a picture that looks like this:
Then I ask them questions like, "Let's say we're on the edge of the circle and the x-coordinate is 4.3. What's the y-coordinate?" And from there it's a pretty quick jump to the equation of a circle. Then we move circles around, use the Pythagorean Theorem on then. By the time we're done, we define the unit circle and offer its equation.
That's great, but what's really important about that to me is that we've gotten in the habit of seeing circles as made up of right triangles. That gets us off to the races for developing the definitions of sin and cos. We lose some of our momentum when we spend a day getting used to rotation terminology, and we need to review the special right triangles. I'd like to tighten up my transition from the opening activity to this one:
Intro to Trig P5 Day Four
Intro to Trig Day 5 Worksheet
Honestly, I can see myself combining this with Kate or Riley's intro activity. The purpose of all of these introductions is the same -- immersing students in a single, concrete conceptual model that students can return back to throughout trigonometry. My introduction is just my attempt to get right triangles in the mix from the very beginning.
Here's the powerpoint file from the beginning of the unit on equations of circles:
Equations of Circles Opening Activity