I'm currently investigating the following system of equations
$$ t(x)=\sin(x)+c\\ s(x)=q(a;c)\cdot \sin(x+a) $$
Where the goal is to have them 'Touch' as shown here https://www.desmos.com/calculator/fnekyssei2
Touch is defined as $\exists x. ( t(x) = s(x) )\ \wedge\ ( t'(x) = s'(x) )$
This means that this system reduces to ( for some $x$ which is the touching point )
$$ \sin(x) + c = q(a;c)\cdot \sin(x+a)\\ \cos(x) = q(a;c)\cdot \cos(x+a) $$
Now I want to solve this for $q(a;c)$, but I'm unsure how to continue.
For $c=1$ it seems like the solutation for $q(a;1)$ is $2 \cdot \cos(a)$
so you are looking for $(q,x_0)$ that satisfy : $$ \left\{ \begin{array}{l} \sin(x_0)+c = q \sin(x_0+a) \\ \cos(x_0) = q \cos(x_0+a) \end{array} \right. \tag{1} $$ with given $a$ and $c$. you can make this to something like a 2 by 2 linear system for $\sin(x_0)$ and $\cos(x_0)$. $$ \left\{ \begin{array}{l} (q\cos(a)-1) \sin(x_0)+ (q \sin(a)) \cos(x_0) = c \\ (-q \sin(a)) \sin(x_0) +(q\cos(a)-1) \cos(x_0) = 0 \end{array} \right. $$ so it becomes: $$ \rightarrow \left(\begin{array}{rr} q\cos(a)-1 & q \sin(a) \\ -q \sin(a) & q\cos(a)-1 \end{array}\right) \left(\begin{array}{r} \sin(x_0) \\ \cos(x_0) \end{array}\right) = \left(\begin{array}{r} c \\ 0 \end{array}\right) \tag{2} $$ to solve this we should calculate the determinant of coefficients matrix $d := det = q^2 -2q\cos(a)+1$. then $(2)$ becomes: $$ \left\{ \begin{array}{l} \sin(x_0) = \frac{qc\cos(a)-c}{d} \\ \cos(x_0) = \frac{-qc \sin(a)}{d} \end{array} \right. \tag{3} $$ next in order for $(3)$ to have solution we should have : $\sin^2(x_0) + \cos^2(x_0)=1$. this gives us: $$ \frac{q^2c^2-2qc^2\cos(a)-c^2}{(q^2-2q\cos(a)+1)^2} = \frac{c^2 d}{d^2}=1 $$ because from $(2)$ $d\neq 0$, we find that $\underline{d=c^2}$. solving this for $q$ gives : $$ q = \cos(a) \pm \sqrt{\cos^2(a)-1+c^2} \tag{4} $$ and $x_0$ can be found with eather of $(3)$. in order to have a real-valued answer for $q$ and $x_0$ there are some conditions we should consider: $$ \left\{ \begin{array}{l} \text{from } (2) \rightarrow d \neq 0 \\ \text{from } (4) \rightarrow c^2>1-\cos^2(a) \\ \text{from } (3) \rightarrow -1 \leq q \cos(a)-1 \leq1 \\ \text{from } (3) \rightarrow -1 \leq q \sin(a) \leq1 \end{array} \right. $$ see this https://www.desmos.com/calculator/h4dhomrln0. in this link consider that changes in $a$ and $c$ may fail the touching because the above conditions may not satisfied.