Desmos construction : https://www.desmos.com/calculator/uul5rbl6gd
Note : in the case of non rotated curves, the coordinates of the point of tangency can be " read off" on the equation of the tangent ( point-slope formula ... ) , but it's not the case here; hence my question
Let $C$ be the curve defined by : $f(x)= 3 \cos(x/3)$.
Let $(a, f(a))$ be a moving point on $C$.
Let $C'$ be the image of $C$ under a counterclockwise rotation of $R$ radians; namely, $C'$ is defined by :
$ Y(x,y) - f( X(x,y)) = 0$ ( or, equivalently : $ Y(x,y) = f( X(x,y)))$,
with ( by the rotation of axis formula)
$X(x,y) = x\cos(R) + y \ sin(R)$
and
$Y(x,y)= y \cos(R) - x \sin(R) $.
Let
$$X(x,y) - f(a) = f'(a) ( X(x,y) -a )$$
be a moving straight line constantly tangent to $C'$.
My question is how to define the ( moving) point of tangency ( or the moving point of contact between $C'$ and its moving tangent) , in terms of $a$.
Note : The attenpt at defining the point of tangency as $( X(a, f(a)), Y(a, f(a)) $ does not work, because this last point follows the image of the curve rotated in the wrong sense.