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I'm having trouble figuring out how to convert this equation to Cartesian coordinates. Sorry if I didn't format my question correctly, this is my first time using this site. Any help would be appreciated! $$r = \frac{1}{2\cos(\theta)+3\sin(\theta)}$$

Abel
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user1781500
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3 Answers3

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If we multiply both sides with $2\cos(\theta)+3\sin(\theta)$ we obtain $2r\cos(\theta)+3r\sin(\theta) = 1$. Since $x=r\cos(\theta)$ and $y=r\sin(\theta)$ we can rewrite this to $2x+3y=1$.

Abel
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  • Thanks! The problem wants me to actually find the coordinates and describe the resulting curve. Would I just put that equation into slope intercept form? – user1781500 May 02 '13 at 16:06
  • I'm not sure if the problem even requires that, since 'describe the resulting curve' is a little vague. It couldn't hurt I guess. – Abel May 02 '13 at 16:13
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If $$r = \frac{1}{2\cos(\theta)+3\sin(\theta)}$$

then

$$2r\cos(\theta) + 3r\sin(\theta) = 1$$

Now remember what $x$ and $y$ are in terms of $r$ and $\sin(\theta), \cos(\theta)$.

Christopher
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To go between polar coordinates and Cartesian coordinates, you can use that $$\begin{align} x &= r\cos(\theta) \\ y &= r\sin(\theta) \\ r^2 &= x^2 + y^2 \end{align} $$ So you can start by rewriting your equation as $$ r[2\cos(\theta) + 3\sin(\theta)] = 1. $$

Thomas
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