The unit circle
A circle of radius 1. Drag the point around it. The horizontal leg is cos θ, the vertical leg is sin θ, and the segment on the vertical line at x = 1 is tan θ. On the right, the sine wave unrolls as the angle grows.
What's going on
In a right triangle, sine is opposite over hypotenuse. Put the triangle inside a circle of radius 1 and the hypotenuse is 1, so sine is simply the height of the point, and cosine is simply how far right it is. That is the whole trick: it turns two ratios into two coordinates, and coordinates make sense for any angle, including ones no triangle could have. Past 90° the point keeps going round, so sin 150° is the height of a point in the second quadrant, which is the same height as sin 30°.
Pythagoras on the triangle says (cos θ)² + (sin θ)² = 1². That is the identity sin²θ + cos²θ = 1, and now it is obvious rather than memorised. Tangent is sine over cosine, and the orange segment shows why it is also a length: the big triangle with base 1 is similar to the small one, so its height is sin θ / cos θ. When cos θ hits zero the segment shoots off to infinity, which is why tan 90° does not exist.
Radians measure the angle by the length of arc it cuts on this circle. A full turn is the circumference, 2π, so 180° is π and 60° is π/3. The unrolled sine wave makes the point: the horizontal axis on the right is literally the distance travelled around the circle.