Ok, obviously, sine is a function. But what I really want to know is what is going on under the hood of sine. What equations and algorithms make up sine? I have searched the web, but I am not sure what to search for and haven't come up with anything useful. I have grepped math.h, but to no avail. Any help is much appreciated.

http://en.wikipedia.org/wiki/Sine#Right_triangle_definitions

lol just incase you don't like his wikipedia link, ill just sumarise it quickly.

 

the sine of an angle, most easily, is the ratio of the side that is opposite the angle, to the side that is the longest, or the hypotenuse. We can extend this to angles other than those in a right angled triangles by enscribing the triangle in a circle with radius one, which is a tiny bit more complex.

Yeah sine is no more than a ratio of sides. It's particularly usefull since it's value corresponds to the amplitude of a wave at various angles where 1 wavelength = 360 degrees = 2pi.

I also have a question about sin, which is when we input sin(90) ( or sin of any degree measure) into our calculator then what does the calculator do to come up with the answer (in this case 1)?

Does it use unit circle to determine the value of it? or is there any algorithm?

 

if sin is as simple as described above, then why we even call it a function and make it sound like some sort of higher math?

Well, firstly because it's clearly a function. It's not "higher math", rather a fact. It's given a special name since there's a whole load of things that you can do with the trigonometric functions that you just can't do with a lot of other functions. For example, Fourier series are completely reliant on the idea of expanding a function in terms of sines and cosines, and this is in turn immensely important in the field of partial differential equations.

 

In the case of a calculator, I would imagine that there is certainly some sort of algorithm used. A Taylor expansion can give very accurate results within the range [math][-\pi, \pi][/math] using very few terms, and since sin is periodic, this seems to effectively solve the problem. However, I don't know for sure that this is the method that they use.

Not to mention, we call it sine so that we don't have to explain it over and over again -- 'sine' is a much shorter way of saying what everyone has agreed to be the defintion. Kind of like why we agree to call a number or variable 'squared' instead of saying 'multiply it by itself' over and over again. As dave alluded to, calling it a 'function' implies certain properties about its behavior, all of which sine fulfills. Whether you want to call it higher math is I suppose is up to you personally, but f(x)=x is also a function, heck, f(x)=4 is too. They are functions because they fulfill all the requirements to be called functions.

Also, you should know that sine function is an odd function, while cosine is an even function. That's why they have their own special name to define the functions.