Why do trigonometric functions have reciprocals?

I've gotten to the point in my trigonometry studies where I am looking at Cosecant and Sine, and I see that Cosecant and Sine are reciprocals of each other. Now this means that multiplied together produce 1 (on the Unit Circle) and I'm assuming everywhere else.

Now why is their relationship a reciprocal one? I see as Sine gets smaller, Cosecant (and Cotangent) get larger. Is there a website that specifically describes why the changing of one causes a change in the other one (outside of the obvious). I drew pictures of them, so I can visually see the change, but I don't understand why the numbers change the way they do.

Does that make sense?