it's a little hard to explain every little thing if you don't know lots of math. I'll give you the basics:
given a right triangle (a triangle which the max angle has 90º or Pi/2 radians), the sine of the angle x is defined as the quocient between the oposite side and the hipotenuse (side in front of the right angle).
With this definition, we can calculate some values of sine for some angles:
1) 30º = Pi/6: imagine you have an equilateral triangle (all of the three sides have the same length). The three internal angles of this triangle have 60º (Pi/3 radians). Now, draw a bissetrix of one of these angles and it will divide your triangle into two right triangles. To calculate the sine of 30º, take the angle divided by the bissetrix (60/2=30) and make oposite side over hipotenuse: l/2/l = 1/2.
2) 60º = Pi/3: take the same divided triangle of above. Sin 60º = length of the bissetrix (use pithagoras theorem to find that it's l*sqrt(3)/2) over hipotenuse (l): so sin 60º = sqrt(3)/2;
3) 45º = Pi/4: take a right triangle with internal angles equal to 45º, 45º and 90º. Using pithagoras theorem, we find that the hipotenuse is equal to the others sizes plus sqrt(2). So, Sin 45º = l/l*sqrt(2) = 1/sqrt(2) = sqrt(2)/2.
there are some formulas involving sine and cossine:
sin(2x) = 2sin(x)cos(x);
sin(x + y) = sin(x)cos
+ sin
cos(x);
sin(x - y) = sin(x)cos
+ sin
cos(x);
to find sin(90º=Pi/2), just do it:
sin 90º = sin (2*45º) = 2sin(45º)cos(45º) = 2*sqrt(2)/2 * sqrt(2)/2 = 1; (obs: the cossine of 45º = sqrt(2)/2);
but, how computers calculate, for example, Sin(18*Pi/107), Sin(0.235), Sin(100), etc?
When, every funtion in the world can be approximated to a polinomial when the value you wanna calculate is near some known value. The formula is that:
:homestar: f(x) = f(a) + f’(a)*(x-a) + f’’(a)*(x-a)²/2! + f’’’(a)*(x-a)³/3! + f’’’’(a)*(x-a)^4/4! + ...
if x->a; (f’ is the first derivative, f’’ is the second derivative, etc...)
if we apply this aproximation for a point x near to 0 (a=0) using the sine funtion, we have:
sin(x) = sin(0) + cos(0)*x - sin(0)*x²/2 - cos(0)*x³/6 + sin(0)*x^4/24 + ...
but sin(0) = 0 and cos(0) = 1 (just use the formulas above to get this).
So, if x is near to 0, sin(x) = x - x³/6 + ...
computers use this formula to calculate sine, another one to calculate cossine and another to calculate the tangent.
:homestar: But, if the angle x is not near 0. If x=61*Pi/3, for example?
Well, take the formula sin(x+y) = sin(x)cos
+sin
cos(x) to find that
sin(61Pi/3) = sin(20Pi + Pi/3) = sin(20Pi)cos(Pi/3) + sin(Pi/3)cos(20Pi) = Sin(Pi/3), because Sin(20Pi) = 0 and Cos(20Pi) = 1.
I hope you have undestood something... There's a lot more to explain, but these are the basics.
Go to this site:
http://mathworld.wolfram.com/Sine.html
