They're related as follows: The reason you get nice relationships for cos(theta) = adj/hyp is due to pythagreon's theorem. a^2 + b^2 = c^2
take c = 1
a^2 + b^2 = 1, let a = sin x, b= cos x (from a definition below)
sin^2(x) + cos^2(x) = 1, so that leaves the sides of your triangle as the base cos(x) and the height sin(x) since your triangle can be inscribed in a circle. And in the image below, I've shown thats how they arise naturally. You can see how the rest falls out .
P.S. I had to do some digging for this. As I read this post I realized that no teacher in my history of highschool, college, or grad school has ever given a good definition of trigonomic functions to me. I had all the pieces but I needed to really think about how they all fit together. I understand now why they are periodic functions, and its all due to our friend Pythagoreous.