We use two special right triangles - 45-45-90 and 30-60-90 - and draw them in each quadrant to create the coordinates on the unit circle.
Let's start with what the special right triangles look like.
We make the hypotenuse (radius) 1 so that each triangle will fit within the same circle.
Now, take the 45-degree reference angle triangle and draw it in each quadrant.
The adjacent leg represents the x-value and the opposite leg represents the y-value. This creates the four coordinates for the 45-45-90 triangle (#pi/4, (3pi)/4, (5pi)/4, (7pi)/4#).
Let's move on to the 30-degree reference angle triangles.
This produces the coordinates for #pi/6, (5pi)/6, (7pi)/6, and (11pi)/6#.
Finally, let's draw the 60-degree reference angle triangles.
This produces the coordinates for #pi/3, (2pi)/3, (4pi)/3, and (5pi)/3#.