Question

find all vertical and horizontal tangent lines on r=4cos(3θ)

Answer 1

for horizontal tangent lines, you need to find theta values where slope is zero
differentiate r and set equal to 0
r' = -12sin(3x) = 0
sin(3x) = 0
sin is 0 at 0,pi,2pi...
3x=0 -->x=0
3x =pi --> x=pi/3
3x=2pi -->x=2pi/3
...
theta = 0,pi/3,2pi/3,...