\cos(2\pi x), \cos(4\pi x), \cos(8\pi x), \cos(16\pi x), and \cos(32\pi

Motalli21

Motalli21

Answered question

2022-01-26

cos(2πx),cos(4πx),cos(8πx),cos(16πx), and cos(32πx) are all nonpositive. What is the smallest positive x?

Answer & Explanation

kumewekwah0

kumewekwah0

Beginner2022-01-27Added 14 answers

Assuming x(0,1), let the binary representation of x be 0.a1a2a32
The condition that cos(2πx)0 is equivalent to that (a1,a2)=(0,1) or (1,0) (or technically x=34, but note that 34=0.101112)
Similarly the rest are equivalent to that (a2,a3),(a3,a4),,(a5,a6){(0,1),(1,0)}, and one may see that the smallest such x  is  0.0101012  or  2164.

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