Im assuming you have a line connecting 2 points... A & B

Now the thing that sin and cosine help out with is the fact that when you rotate
a point about the origin (think of a circles circumfrence) it always stays the same distance from the center point. By using this we can work out what the Adjacent & Opp sides are, now Look at this picture



Now if R is the angle we used to find the rotation then X Axis is the Adjacent side and the Y Axis is the Opposing side.

So say we want to find a Point F which lies on a Circle of 10units, and it is at 33 Degrees, we simple plug it into this thing and find

Adjacent = Radius * Cos(33)
Opposing = Radius * Sin(33)
Thus
X = Radius * Cos(33)
Y = Radius * Sin(33)

And by replacing 33 with Angle we can draw a circle using
Radius = 100
for angle = 0 to 360 do
{
X = Radius * Cos(Angle)
Y = Radius * Sin(Angle)
PutPixel(x,y);
}

Now this is the 1 way it goes, so if we know a unit is Facing 66 Degrees we can work out its Facing Normal by using the top
-----------------------
Normal.X = Cos(FacingAngle)
Normal.Y = Sin(FacingAngle)

And now if the unit moved 5m then we can find its new location by
Speed = 5m
Position = Position + Normal * Speed


--------------------------
Thats the easy part, the difficult part is going backwards ie, from the normal to the angle.

And anyways its beyond the scope of this

Visualise this your standing at a point and your looking in a direction so increasing the Z,X will not move you proporly so you define the speed you want to move in , thats the radius of the cirle now you wanna know the increase that should happen in the x , z so you draw line from the point your standing in to the radius with angle 0 thats the z then draw a cord to connect the two radiis thats the x now you have a right angle triangle so x is the oppiste and z is the adjacent if you rembemer from your math lessons

SOH Sin(Angle) = Oppiste / Hypotense
CAH Cos(Angle) = Adjacent / Hypotense

the angle is the angle we wanna move with and the hypotnese is the speed we moved with so by some Algebra we could do this

oppiste = Sin(Angle) * Hypotense
Adjacent = Cos(Angle) * Hypotense

Which Simplifies to
X = Sin(Angle) * Hypotense
Z = Cos(Angle) * Hypotense

So Now We have the two coord we can compute the point on the circumfrunce we want to move to (X,Z) , any 2 axis can be seen as 2d so you can do this with x,z y,x y,z how ever you like and with some modification this can be used to compute Rotation , Intersection etc.. if your intersted in how is it actually computed you just get the sin,cos of one number and by Cross-Muliplication you can find any other number.
Good Luck :D

Hmm a tough question. Uhhm, your calculator comes up with an approximation of the ratio by doing something fancy with fractions I believe.

Click Here AND Click Here

Now have a look there at the graphics and you might begin to understand what AP was talking about with the unit circle.

Now, when you see Sin45, have a look at the second link I gave you - Sin45 = 1/sqrt(2) = .707 do you see where 45 is around about on the graph? The y-coordinate of that is at about .707

Another example, by looking at that same graph can you see what Sin90 will be? Sin180? Sin270?

Hrrm this probably isn't very clear and might seem unrelated to your question but I hope it led you into a little more of an understanding. You just have to fiddle around yourself and you will start to comprehend I guess.

Merry Christmas, take it easy
-Jemburula

Sin/Tan/Cos are all related and are all infact RATIOS
------------------------------------------------------
When That idiot founded the pythagoras theorem (a^2 = b^2 + c^2) which works on 90 degree angles they took this further and built a table of ratios saying.

Opp/Hyp = Sin(Theta)

Where Theta is the angle in a triangle, then they went and did this for all angles (with a bit of logic).

Now since we know the ratio(sin look up table) and the Angle Theta we can work out the Oppor Hyp if we know either 2, so the connection is

Sin(Theta) = Opp/Hyp

So now we know that Theta is 45 Degrees, look it up (or use the calc) and you are left with
(numbers are fictional)
12 = Opp/8

And so in effect SIN/COS/TAN are all derived from the Pythagorean Theorem

Now Cos is essentially the same as sin in that sin(x) = cos(x-90)

And to look at some graphs of the COS/SIN table look at
http://www.mathsrevision.net/gcse/sin_cos_tan.php

ok this is short and simple if you understand the pythagoran theorm you will understand this someone brought a right angle triangle and kept changing the lenght of each side they found that they have a ratio between them so they mesured one right angle triangles sides and angles and began recording them into a table they called the angle they were trying to get Theta and the side oppisite to it well OPPISITE and the side oppisite to the Right angle hypotense and the side thats left adjacent , they called the ratio between the oppisite and the Hypotense Sine and the ratio between the Adjacent and the hypotense Cosine and the ratio between the oppisite and the adjacent Tan and they started using the tables of the ratios since then.
Hope This Helps

EDIT:
here is a guess if i made a calc i would do it like this
i would get a right angle triangle with an angle thats lets say 40 then get the mesure of the oppisite side and the hypotense then divide them to get the ratio and that would be sin 40(0.642) then when you wanna get the sin of another number i would do a cross multiplication like this (0.642 * Num) / 40
that would get the sin of the other number.