These are sums of cos(w*j), where j belongs to a set Ai. Sets Ai are of the form
{ x, M(2*x), M(4*x),..., M(32768)}
(see page of Literka
Part 65537-1).
We split set Ai into two sets Ci and Di of 8 elements each
Ci={x, M(4*x), M(16*x), M(64*x), M(256*x), M(1024*x), M(4096*x), M(16384*x)}
Di={M(2*x), M(8*x), M(32*x), M(128*x), M(512*x), M(2048*x), M(8192*x), M(32768*x)}
Let us notice that our construction is very similar to the construction of the page
Angle257. Part VI.
Let us define
We have
In a known way, repeated many times, we can prove, for example, that
Hence c1 and d1 are the roots of the equation
Other equations can be derived the same way. Below there is an applet showing all formulas. Later we'll
regard ci and di as known values.
In the following applet use applet's scrollbar to see whole formulas.
Software that you really need.