The unit circle
A right triangle only has room for an acute angle. What could the cosine of 140 degrees, or the sine of a negative angle, possibly mean?
Where you are 8 inside
What this area is about
Put a circle of radius 1 at the origin. Every angle in standard position, of any size or sign, has a terminal side that meets the circle at one point (x, y), and that point defines \cos\theta = x and \sin\theta = y. For acute angles this agrees with the right triangle ratios; for every other angle it gives a value where the triangle gave none.
How these fit together
The definition. Trig functions on the unit circle makes cosine and sine coordinates, and tangent the slope of the terminal side. Secant, cosecant and cotangent on the unit circle adds the other three and finds where a zero coordinate leaves no value, such as \tan\theta at \displaystyle \theta = \frac{\pi}{2}.
Structure of the circle. Symmetry and the even-odd identities and periodicity (2\pi for sine and cosine, \pi for tangent) relate the values at different angles. Sine, cosine and tangent as functions of a real number reads a radian input as distance along the circle.
Procedures. Signs by quadrant turns two signs into a quadrant. Reference angles reduce any angle off the axes to an acute one, up to sign. Exact values on the unit circle collects the exact values at multiples of \displaystyle \frac{\pi}{6} and \displaystyle \frac{\pi}{4}.
A path through it
The coordinate definition first: every other lesson here builds on it. Reference angles and the reciprocal functions come before exact values, which need both. Symmetry, periodicity, signs by quadrant and the real-number view follow the definition.
Next come the graphs of sine and cosine and trig equations.