How do you find the equation of the tangent line to the polar curve #r=3+8sin(theta)# at #theta=pi/6# ?

1 Answer
Sep 21, 2014

The equation of the tangent line is

#y-7/2={11sqrt{3}}/5(x-{7sqrt{3}}/2)#

Explanation:

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)#