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A year ago, I posted this problem on [MSE]. After a number of edits, I have arrived at the following more general problem (suggested by Hjalmar Rosengren; see the comments below).

For which rational values of c and d are the numbers sin(π⋅c),sin(π⋅d) and 1 linearly dependent over Q?

Notes:

(1) The original question I had asked boiled down to understanding a very special case:

Question 1: If d=118 and 0<c<12, do we have such a linear dependence only if c=118 or c=1/6?

(2) Niven's theorem states that if 0≤c≤1/2, then sin(π⋅c) is rational only if c∈{0,1/6,1/2}. So in some sense I am asking about a generalization of Niven's theorem:

Question 2 If we restrict to rational values 0<c,d<1/2 and demand c,d≠1/6, do we achieve such a rational dependence only if c=d?

GH from MO
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math110
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  • why [on hold]? Thank you – math110 Jan 15 '15 at 10:31
  • such this post,http://mathoverflow.net/questions/130319/show-that-this-ratio-of-factorials-is-always-an-integer I know this is AME(USA) problem,But don't hold,and my problem A class of transcendental trigonometric equations integer solution of such a situation，Related to the this general a+bsinA=c+dsinB problem?So far I have not seen such a study of this problem this paper – math110 Jan 15 '15 at 10:47
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    You can more generally ask for which rational values of c and d the numbers sin(πc), sin(πd) and 1 are linearly dependent over Q. If you can solve this for d=1/18 you are done. The case d=0 is known as Niven's theorem. Maybe you should try rephrasing your question in more general terms. – Hjalmar Rosengren Jan 15 '15 at 14:45
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    Since you have performed major surgery on this question, I went in and edited further to what I hope is a pretty stable state. Please go ahead and insert the reference to the M.SE question, and perform any other minor edits (e.g., a colon after Question 2), and let's see what reaction this now gets (i.e., after one more edit, let's leave it alone for a while). – Todd Trimble Jan 16 '15 at 08:52
  • @ToddTrimble,Thank you for help edit it,and seem it is beatifull.+1 – math110 Jan 16 '15 at 09:42
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    Morris Newman found, many years ago, all solutions in rational a,b,c to sinπasinπb=c (and later I extended this to products of 3 and 4 sines). The methods involve writing it as a vanishing sum of roots of unity (and there's a paper by Conway and Jones about that). I suspect one should be able to cobble together an answer out of those sources and techniques. – Gerry Myerson Jan 16 '15 at 16:11
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    The sine values for c=1/10 and d=3/10 differ by 1/2, so the answer to question 2 is no. – Zack Wolske Jan 16 '15 at 19:24

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