Draw a line on a piece of paper at ANY angle you want.
Pick which end to start from and draw a perfect horizontal line in the same direction as the other end of the line. Now draw a perfect vertical line up or down to the other end. You end up with a right triangle (90 degrees) and guess what? The radius (hypotenuse) is longer than or equal to one other side. Equal to would mean it was drawn horizontally or vertically.
SIN and COS times the radius are the horizontal and vertical sides of that triangle. So Hypotenuse = SQR(horiz ^ 2 + vert ^ 2). If the line is horizontal then there is no vertical value. Same way with vertical.
If you draw EVERY possible right triangle intersection point with the SAME length hypotenuse, you create a PERFECT circle!
The GREAT thing is that now we don't have to have the freaking book to tell you the ratios for any given angle. It is built into your PC.
We used "slide rules" at CMU! Look that up....