It is a very well known result of Gauss that a regular polygon with 17 sides (heptadecagon) can be constructed by a ruler and a compass
(see a page of Literka Construction of a regular polygon with 17 sides.)
This means that the values cos(2*k*pi/17) can be
expressed in terms of square roots and arithmetic operations of integers. Formula for cos(pi/17) and a proof for this formula
can be found on the page of
Satoshi Hoshino. Literka has managed to find a general
formula for all values cos(2*k*pi/17). This formula looks like this:
where
is a sequence of signs i. e., a sequence of "one" or "minus one". Of course, we have 8 such sequences. For each sequence of this form there is a corresponding one
number of the form cos(2*k*pi/17).
For example,
for a sequence {1,1,1} the corresponding number is cos(2*pi/17). Hence the number cos(2*pi/17) is equal
Proceeding the same way, for a sequence {1,1,-1}, we get a formula for cos(8*pi/17):
For a sequence {1,-1,1} we receive a formula for cos(4*pi/17):
A sequence {1,-1,-1} gives us a formula for cos(16*pi/17):
A sequence {-1,1,1} corresponds to a formula for cos(6*pi/17):
And a sequence {-1,1,-1} gives us a formula for cos(10*pi/17):
For a sequence {-1,-1,1} we get a formula for cos(12*pi/17):
Finally, for a sequence {-1,-1,-1} we get a formula for cos(14*pi/17):
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