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I know the definition of sine and cosine, but how were these formulas originally invented? I mean, how did people derive the power series for sine and cosine for the first time?

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I know the definition of sine and cosine, but how were these formulas originally invented? I mean, how did people derive the power series for sine and cosine for the first time?

- #2

HallsofIvy

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You can also define sine and cosine by:

"y= sin(x) is the function satisfying y"= -y for all x, with y(0)= 0, y'(0)= 1 and y= cos(x) is the function satisfying y"= -y for all x, with y(0)= 1, y'(0)= 0" and, personally, I prefer that definition. From that the Taylor's series about x= 0 immediately follow.

- #3

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(cos(x+dx)-cos(x))/dx = (cos(x)cos(dx)-sin(x)sin(dx)-cos(x))/dx = sin(x)sin(dx)/dx --> sin(x)

? (I'm sorry this is so ugly, I hope you understand what I mean. Is TeX broken?)

The only problem is that I have to argue that sin(dx) goes to zero linearly, but that seems to make sense geometrically.

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Edit:

Yes, LaTeX is broken at the moment :(

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HallsofIvy

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So we have (1/2)cos(t)sin(t)<= (1/2)t<= (1/2)sin(t)/cos(t). Multiplying through by 2/sin(x), we have cos(t)<= t/sin(t)<= 1/cos(t) which, inverting, gives 1/cos(t)<= sin(t)/sin(t)<= cos(t). Since cos(t) is continuous and cos(0)= 1, taking the limit as t goes to 0, lim sin(t)/t= 1.

We also will need sin(x+y)= sin(x)cos(y)+ cos(x)sin(y) and sin(x-y)= sin(x)cos(y)- cos(x)sin(y). Adding those sin(x+y)- sin(x- y)= 2cos(x)sin(y). In particular, if we take A= x+y and B= x-y, then x= (A+ B)/2 and y= (A-B)/2 so that sin(A)- sin(B)= 2 cos((A+B)/2)sin((A-B)/2).

Now we can write sin(x+h)- sin(x)= 2cos((2x+h)/2)sin(h/2)= 2 cos(x+ h/2)sin(h/2)

Then [sin(x+h)- sin(x)]/h= 2 cos(x+h/2)sin(h/2)/h= cos(x+h/2)sin(h/2)/(h/2).

lim(h->0) [sin(x+h)- sin(x)]/y= lim(h->0) cos(x+ h/2) lim(h->0) sin(h/2)/(h/2). cos(x+ h/2) goes to cos(x) and sin(h/2)/(h/2) goes to 1 so that limit is cos(x): the derivative of sin(x) is cos(x).

- #6

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sin(x)=x-(x^3)/3+...

does this not go to zero linearly ? I.e. lim(x->0) sin(x) = lim(x->0) x.

Anyway, it is a nice proof and a nice way to derive sine and cosine that you have told me. What about before people knew Taylor series? I just tried to do some geometric approximations (approximating an arc by a straight line), and I got

cos(x)=1-1/2*x^2

but I could not get much further because the equations got too complicated ...

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