Why our "definitions" work so well? The one thing common to all mathematical and physics concepts is that we 'assume' and 'define' various 'things'. I've not studied any branch of mathematics or physics in a very rigorous fashion, so I cannot give solid examples to substantiate my argument but I can surely list some of the examples which are elementary. 
For example when we consider trigonometric functions of real numbers, what we do is define them. In this case we use unit circle definition, which according to me is arbitrary. We could define trigonometric functions by considering sine of a real number as twice of the ordinate of corresponding point on unit circle, cosine of a real number as one-half of the abscissa of the corresponding point on unit circle and tangent of a real number as square of ratio of sine of that number to the cosine of that number. This was an arbitrary definition and so is the definition which is used (unit circle definition).
So definition is purely arbitrary with no logic (according to me). But the problem I have is not with the definition we give but with the fact that it works so well in all possible applications which a person might not be aware while constructing the definition. So, for example, when we have to calculate work done on a particle at an angle, we blindly apply the formula wherein we substitute the value of cosine of the angle as per the definition, and receive the answer. The surprising thing is that what we receive by applying our "arbitrary definition" completely agrees with what actually happen. Like it is indeed easier to move a thing tied to a string making an acute angle with the direction of motion, than by applying force parallel to it.
Another example is the way vector addition is defined, which is also arbitrary. But in actual physical situations we find that our definition is correct. For example if we are rowing a boat in direction perpendicular to the motion of river, our resultant direction of motion is indeed somewhere between the two directions. So why our definition works which is purely arbitrary.
One possible explanation which came to my mind is that it must be looking at various applications, the definition is constructed. But, then again we can not possibly include every possible application of that concept, but mostly these definitions are quite general when it comes to applications.
So what is the reason behind this all. Any help is appreciated. I tried to explain myself best. Sorry for any discomfort.
Thank you.
 A: I am reminded of defensive play by Willy Mays referred to as The Catch. Basically, he made an over-the-shoulder, running catch in deep center field, then immediately threw the ball to second base to stop the man on second base from scoring. 
If you don't understand the game of baseball, you might think, "Big deal. He caught a ball and threw it to second base." Catching the ball over your shoulder while running at full speed to where the is going to be requires buckets-full of skill. He then had to get the ball out of his glove, into his hand, spin, and use the momentum of that spin to throw the ball to second base fast enough and accurately enough to make the runner not want to try for home plate. 
You might wonder how Willie made that play look so easy. By the time he had to make that play, he knew exactly what to do and how to do it because he had worked out all of the problems in hundreds of practice sessions. Then, just to keep sharp, he had probably practiced it a few hundred times more.
The definitions of the trigonometric functions are like that. Believe me, there is nothing arbitrary about those definitions. You just don't see the years that mathematicians spent making it look as simple as they could.
If you want to really understand the definitions. You will have to find out about their history. Where did they come from? What problems did they solve? Simplicity of a definition is a result of a lot of hard work, failures and successes. There is nothing arbitrary about it.
