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 $\mathbb{Q}$? 

Notes: 

(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 $c = d$? 





[Niven's Theorem]:http://mathworld.wolfram.com/NivensTheorem.html