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 $c = d$?