We started off introducing the concept of radian angle measure. Take an arc length of s, on a circluar sector of radius r, to get the angle θ, in radians. Of course most students are more familiar with degree measure and so you should be able to convert back and forth between radians and degrees. However it is also important to be able to count out units on the unit circle in radians alone.
From there we figured the revolutions per time unit, linear speed, or angular speed of a point on a spinning disk a distance of r, from the center. We should be able to do problems like this using unit analysis as conversion problems with just a few basic facts such as the number of radians in one revolution or the amount of arc length per radian.
We took the circle and placed it on the x-y grid, centered it at the origin, gave it a radius of 1, and called it the Unit Circle.
Given a point (a, b) on the unit circle at an angle of t measured counter-clockwise from the positive x-axis, i.e. P(t)=(a, b), we practiced moving about the unit circle and finding other points such as P(π/2 - t) and P(3π/2 - t), etc.
Next we defined the trig functions in terms of the unit circle. Start at the point (1, 0). Move up a distance of t along a vertical line, wrap the line around the unit circle. Look at the point you land on. The x-coordinate of the point is the cosine of t (abbreviated cos t) and the y-coordinate of the point is the sine of t (abbreviated sin t). Tangent (abbrv. tan), cotantent (abbrv. cot), secant (abbrv. sec), and cosecant (abbrv. csc) can all be thought of in terms of sine and cosine or in terms of ratios of the coordinates of the point that you land on.
We discovered that there are certain values we can find exactly on the unit circle and thus we know the exact values of the trig functions there. A little bit of geometry is all it takes and we know the trig functions at π/6, π/4, π/3, and π/2 and any multiple of those angles.
We dropped a line down from our landing point on the unit circle and drew in the radius to make a right triangle, took that right triangle out of the unit circle and let the radius change from 1 to r in general.
We redefined the trig functions in terms of the sides of a right triangle using the names adjacent, opposite, and hypotenuese where the hypotenuese is the side opposite of the right triangle. Which side is the adjacent and which side is the opposite depends on which acute angle you are looking at. The adjacent side is teh side touching the angle in question adn teh opposie is teh side opposite of the angle in question. Of course we also did several application problems that use right triangles.