In order to find the equation of a line, we need two pieces of information:
#{(1. "Point: " (x_1,y_1)),(2. "Slope: " m):}#
Let us find #(x_1,y_1)#.
Since
#{(x(theta)=rcos theta=(3+8sin theta)cos theta),(y(theta)=rsin theta=(3+8sin theta)sin theta):}#,
#x_1=x(pi/6)=[3+8sin(pi/6)]cos(pi/6)={7sqrt{3}}/2#
#y_1=y(pi/6)=[3+8sin(pi/6)]sin(pi/6)=7/2#
Now, let us find #m#.
By differentiating with respect to theta#,
#{dx}/{d theta}=8cos theta cdot cos theta+(3+8sin theta)cdot(-sin theta)#
#=8(cos^2theta-sin^2theta)-3sin theta#
#=8cos(2theta)-3sin theta#
#{dy}/{d theta}=8cos theta cdot sin theta+(3 + 8sin theta)cdot cos theta#
#=8(2sin theta cos theta)+3cos theta#
#=8sin(2theta)+3cos theta#
So,
#{dy}/{dx}={{dy}/{d theta}}/{{dx}/{d theta}}={8sin(2theta)+3cos theta}/{8cos(2theta)-3sin theta}#
Now, we can find #m#.
#m={dy}/{dx}|_{theta=pi/6}
={8({sqrt{3}}/2)+3({sqrt{3}}/2)}/{8(1/2)-3(1/2)}={11sqrt{3}}/5#
By Point-Slope Form: #y-y_1=m(x-x_1)#,
#y-7/2={11sqrt{3}}/5(x-{7sqrt{3}}/2)#
