calculus

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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
so y-2=0
y=2
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)

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