the unit circle
Right-triangle ratios only make sense for angles between 0 and 90 degrees — you cannot draw a triangle with a 200 degree angle. The unit circle is the device that frees sine and cosine from the triangle and lets them describe every angle, including negative ones and ones bigger than a full turn. It is the single most important picture in trigonometry.
Draw a circle of radius 1 centred at the origin of a coordinate plane. Start at the point (1, 0) and rotate counterclockwise through an angle theta. You land on a point P on the circle, and that point's coordinates ARE the cosine and sine of theta: P = (cos theta, sin theta). So cosine is the horizontal coordinate and sine is the vertical coordinate of the rotating point; tangent is their ratio sin theta / cos theta, the slope of the line from the origin to P. For acute angles this matches the old triangle definitions exactly (the radius is the hypotenuse of length 1), but now any angle works — rotate clockwise for negative angles, keep going past 360 degrees for repeats.
Everything else flows from this picture. The signs of sine and cosine in each quadrant, the reference-angle trick, the periodicity and the wave shape of the graphs, and the famous values at 30, 45, 60, and 90 degrees can all be read straight off the circle. Memorizing the coordinates at those special angles, together with which coordinate is which, is the foundation every later identity rests on.
At theta = 120 degrees the point on the unit circle is in the second quadrant at (-1/2, sqrt(3)/2). Reading off the coordinates: cos 120 degrees = -1/2 and sin 120 degrees = sqrt(3)/2 ≈ 0.866 — note cosine is negative there, exactly as the leftward x-coordinate shows.
On the unit circle the x-coordinate is always the cosine and the y-coordinate is always the sine — no triangle required.
The angle is measured from the positive x-axis and counterclockwise is positive by convention; reversing the convention or starting from a different axis silently swaps sine and cosine and flips signs.