a reference angle
Suppose you need the sine of 150 degrees but only remember the values for small acute angles. A reference angle is the trick that bridges the gap: it pairs every angle, no matter how large or which quadrant it lands in, with a small acute partner whose trig values you already know.
The reference angle is the acute angle between the terminal side of your angle (the rotated radius on the unit circle) and the horizontal x-axis — always measured to the nearest part of the x-axis, never the y-axis. Quadrant by quadrant: in the first quadrant the reference angle is theta itself; in the second it is 180 - theta; in the third it is theta - 180; in the fourth it is 360 - theta (in degrees). Once you have it, the trig value of your big angle equals the trig value of the reference angle, up to a sign you fix from the quadrant. For example, 150 degrees sits in the second quadrant with reference angle 180 - 150 = 30 degrees, and since sine is positive in the second quadrant, sin 150 degrees = + sin 30 degrees = 1/2.
This reduces the whole infinite supply of angles to the handful of acute values worth memorizing, and it is exactly how the unit circle is read in practice. The two things to keep straight are: measure to the x-axis, and attach the sign from the quadrant separately — the reference angle itself is always positive and acute.
Find cos 225 degrees. The angle is in the third quadrant, so its reference angle is 225 - 180 = 45 degrees. Cosine is negative in the third quadrant, so cos 225 degrees = - cos 45 degrees = - sqrt(2)/2 ≈ -0.707.
Strip the angle down to its acute reference partner, look up that value, then restore the quadrant's sign.
A reference angle is always measured to the x-axis, not the y-axis, and is always taken positive and between 0 and 90 degrees; the sign of the original trig value is a separate decision made from the quadrant.