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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?

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    $\begingroup$ Your conversion of the point $(4,\pi/3)$ is wrong. $\endgroup$ – J. M. is a poor mathematician Jul 13 '12 at 9:19
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    $\begingroup$ second point in cartesian coordinate is $(2,2\sqrt 3)$, not $(2,4)$. $\endgroup$ – Aang Jul 13 '12 at 9:20
  • $\begingroup$ @J.M., make that an answer. $\endgroup$ – MvG Jul 13 '12 at 9:22
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    $\begingroup$ @MvG, I'd rather that Vic figure out why avatar's answer is the correct one him/herself. $\endgroup$ – J. M. is a poor mathematician Jul 13 '12 at 9:26
  • $\begingroup$ thanks for pointing it out. I'm trying to do it again. $\endgroup$ – Vic. Jul 13 '12 at 9:29
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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.

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