[seqfan] Re: A327135 vs A026317, sines and cosines at integer arguments

[jean-paul allouche]

Hi

Here is a proof of what I have just said (proving that the
two sequences A327135 and A026317 are actually equal).

Let a(k) := sin(2k+2) - sin(2k)
and b(k) : = cos^2(k) - sin^2(k+1)
(note that k is not necessarily an integer here).

Then
a(n) = sin(2k)cos(2) + sin(2)cos(2k) - sin(2k) = (cos(2)-1) sin(2k) + 
(sin(2)) cos(2k).

Now
b(k) = ((1 + cos(2k)) - (1 - cos(2k+2))/2  = (cos(2k) + cos(2k+2))/2
        = (cos(2k) + cos(2k)cos(2) - sin(2k)sin(2))/2
        = (-sin(2)sin(2k) + (1+cos(2))cos(2k))/2.

Now it is easy to see that

a(k) = C b(k) where C = -2(cos(2)-1)/sin(2)

qed


best wishes
jean-paul




Le 13/11/2019 à 10:13, Richard J. Mathar a écrit :
> A327135 and A026317 consider sign changes of the sine and cosine
> sampled at integer arguments. Is there some proof that they
> are essentially (up to offset) the same?
>
> RJM
>
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