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hmm.....how about this, i dont understand how pi is measured. is it simply use a ruler and stuff? or formulas?
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Well pi is 'defined' as the ratio: circumference/diameter of a circle. So when pi was introduced, using rulers etc was for long the only way to determine the number. But nowadays there exist some more elegent ways, most of them use series. the simplest is probably: pi/4 =1 - 1/3 + 1/5 - 1/7 ...etc. For more see: http://mathworld.wolfram.com/PiFormulas.html
your formula would certainly go to pi (quite fast even; in radians it is proportional to: pi -pi^3/n^2 +O(1/n^4) [this last symbol is use to say: there are more terms but they are of order 1/n^4 or more] well 1/n^2 goes quite fast to 0 for n->inf. So you got pi left. Very nice technique
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hmm....but "sin".....how is sin measured (also tan, cos....)? just by some equipments? how do ppl know that sinx= x-(x^3/3!)+(X^5/5!)-(x^7/7!)...... i fail to understand this..... any simple proof to this??
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well you moght know that the sin and cos are defined as follows: if we take a point on a unitcircle, say (x,y) and we also look at the angle t of a vecor pointing from (0,0) to (x,y) with the x-axis. then sin(t)=y and cos(t)=x and tan(t)=y/x. So the sin and cos are basicly the 2 orthogonal (basicly means: the directions of the 2 projections are perpendicular to each other) projections of a point on the unit circle. Most numerical calculations are done with the series given above. to show where this comes from you need 2 concepts: namely complex numbers and taylors formula. I'll sketch it (BE WARNED! i cant do this without using explicit formula's... i'm sorry..)and you can look the details up on the internet or so
(otherwise it would take me to much time and there is no formula editor on this board
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TAYLORS FORMULA: Basicly says that most functions can be written as follows:
f(x) = 1/(0!)*f(0) + (1/1!)*x*{df(x)/dx|x=0} + 1/(2!)*x^2 {(d/dx)*(df(x)/dx)|x=0} +...
where e.g df(x)/dx means: f(x) differentiated to x. the addition |x=0 means that in this differentiated function we have to take x=0. the (n!) means: "n faculty" = n*(n-1)*(n-2)* ... *1. We define 0!=1.
the sine and cosine functions also can be written in a defferent way: sin(x) = (exp(ix)-exp(-ix))/(2i). Here i is the complex number. this has the property that i^2 = -1. applying taylor's formula to the above expression for sin(x) gives the series given above.
ps i still wonder: how did you get the original formula with which you began this post? Bo
