Find the equation in polar coordinate form for a straight line through the points with polar coordinates (4,0) and (4,π/3). Find the equation in polar coordinate form for a straight line through the points with polar coordinates $(4,0)$ and $(4,π/3)$.
Here's my steps:
1.Write the two points in cartesian coordinates: the two points are $(4,0)$ and $(2,4)$.
2.Find the cartesian equation of the line through $(4,0)$ and $(2,4)$: $$y=-2x+8$$
3.Replace $y$ with $rsin(ø)$ and $x$ with $rcos(ø)$.
so I get $$rsin(ø)=-2rcos(ø)+8$$
->$$r(sin(ø)+2cos(ø))=8$$
The answer in my book is $$rsin(ø+π/3)=2√3$$
which can be written in the form  of $$√3y=4√3-x$$
Can someone point out where I got wrong?

Your Cartesian co-ordinaries are incorrect.
They should be $(4,0)$ and $(2 , 2\sqrt{3})$. Sorry for the bad notation.
Also you then need to use the Rsin$(θ + α)$ form to get it as the book's answer.
