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?
Is there a proof of it somewhere? Like a geometric proof that shows it? I have this image I have been looking at:
http://schools-wikipedia.org/images/226/22690.png
Or am I ahead of myself?
First, there are 3 sides to a right triangle and so 3(2)= 6 ways of "ordering" any two of them: of "abc" the 6 orders of two at at time are "ab", "ac", "ba", "bc", "ca", and "cb". Once you start making fractions of those three sides, since there are two parts (numerator and denominator) there are 6 possible fractions. Calling the "near leg" to an angle , "a", the "far leg", b, and the hypotenuse, "c", three of those fractions, , and , are commonly used.
The other three, , , and are defined pretty much just to be complete. Although, as much as I like Captain Black's response, there are times when you need to divide by "sine", "cosine", or "cotangent" and then "sec", "csc", or "cot" can be convenient, though they are never necessary.