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 $\sin{(\pi\cdot c)},\sin{(\pi\cdot d)}$ and $1$ linearly dependent over $\mathbb{Q}$?


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

Question 1: If $d=\dfrac{1}{18}$ and $0 < c < \frac1{2}$, do we have such a linear dependence only if $c=\dfrac{1}{18}$ or $c = 1/6$?

(2) Niven's theorem states that if $0 \leq c \leq 1/2$, then $\sin (\pi \cdot c)$ is rational only if $c \in \{0, 1/6, 1/2\}$. 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 $c, d \neq 1/6$, do we achieve such a rational dependence only if $c = d$?

  • $\begingroup$ why [on hold]? Thank you $\endgroup$ – math110 Jan 15 '15 at 10:31
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    $\begingroup$ You can more generally ask for which rational values of $c$ and $d$ the numbers $\sin(\pi c)$, $\sin(\pi d)$ and $1$ are linearly dependent over $\mathbb Q$. 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. $\endgroup$ – Hjalmar Rosengren Jan 15 '15 at 14:45
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    $\begingroup$ 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). $\endgroup$ – Todd Trimble Jan 16 '15 at 8:52
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    $\begingroup$ Morris Newman found, many years ago, all solutions in rational $a,b,c$ to $\sin\pi a\sin\pi b=c$ (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. $\endgroup$ – Gerry Myerson Jan 16 '15 at 16:11
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    $\begingroup$ The sine values for $c=1/10$ and $d=3/10$ differ by $1/2$, so the answer to question 2 is no. $\endgroup$ – Zack Wolske Jan 16 '15 at 19:24

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