for what value of r=4sinè have vertical tangent?
In polar coordinates? Theta=90 deg is vertical.
I do not know if you know how to differentiate polar coordinate equations so I changed your equation from polar to rectangular form, I don't know if that is the way you do it.
anyway I got x^2 + y^2 = 4y which is the equation of a circle.
I differentiated that
2x + 2y(dy/dt) = 4dy/dt
dy/dt = -x/(y-2)
for the tangent to be a vertical line, its slope must be undefined.
This happens when the denominator is 0
then substitute that in the equation for x=2
so in Cartesian form the point is (2,2), making an angle of 45º or pi/4 radians.
in polar form (r,theta) it would be (√8,pi/4)