State the exact coordinates of every special angle on the unit circle and use them to evaluate all six trigonometric functions in any quadrant.
The unit circle is the circle of radius 1 centered at the origin. For any angle theta measured from the positive x-axis, the point where the terminal side meets the circle has coordinates (cos(theta), sin(theta)) -- directly, with no scaling, because the radius is exactly 1. This IS the definition of cosine and sine that the rest of trigonometry builds on. The other four trig functions come from these two: tan(theta)=sin(theta)/cos(theta), and their three reciprocals csc=1/sin, sec=1/cos, cot=1/tan.
This connects directly to right-triangle trigonometry from geometry: drop a segment from the circle point straight down to the x-axis, and you get a right triangle with hypotenuse 1, so SOH-CAH-TOA gives the exact same cos(theta) and sin(theta) as the unit circle definition -- the unit circle just extends the idea to every angle, not only acute ones.
A handful of special angles -- 30, 45, and 60 degrees (pi/6, pi/4, pi/3 radians) -- have exact, memorizable coordinates, and every other special angle is just one of these reflected into a different quadrant. The signs in each quadrant follow "All Students Take Calculus" (ASTC): all six functions are positive in QI, only Sine (and cosecant) in QII, only Tangent (and cotangent) in QIII, only Cosine (and secant) in QIV.
Mastery looks like: They state exact coordinates for every special angle and correctly evaluate all six trig functions at any angle by combining the reference angle with the correct quadrant sign.
Common stumbles: Forgetting to apply the quadrant's sign after finding the reference angle's magnitude; mixing up which coordinate is cosine and which is sine.
See it visually
Interactive, spoken-aloud tools that show this idea in motion.