Given #r(theta)# we have
#{
(x(theta)=r(theta)cos(theta)),
(y(theta)=r(theta)sin(theta))
:}#
#
{
(dx/(d theta) = -r(theta)sin(theta)+cos(theta)((dr)/(d theta))),
(dy/(d theta) = r(theta)cos(theta)+sin(theta)((dr)/(d theta)))
:}
#
but
#r(theta) = -8 Sin(theta/4)+ 4 Cos(theta/2)#
then
#
{
(dx/(d theta) = 3 Cos((3 theta)/4) - 5 Cos((5 theta)/4) - Sin(theta/2) - 3 Sin((3 theta)/2)),
(dy/(d theta)=Cos(theta/2) + 3 (Cos((3 theta)/2) + Sin((3 theta)/4)) - 5 Sin((5 theta)/4))
:}
#
but
#(dy)/(dx) = (((dy)/(d theta)))/(((dx)/(d theta)))#
at point #theta_0 = 2 pi/3# we have
#p_0 ={r(theta_0)cos(theta_0),r(theta_0)sin(theta_0)}={1.0,-1.73205}#
and
#m_0 = ((dy)/(dx) )_{theta_0} = -0.57735#
The tangent straight reads
#y = -1.73205 - 0.57735 (x-1)#
