How do you convert polar equations to rectangular equations?

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How do you convert polar equations to rectangular equations?

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Answer 1 · 2015-01-08T09:28:29

To convert an equation given in polar form (in the variables $r$ $r$ and $θ$ $theta$ ) into rectangular form (in $x$ $x$ and $y$ $y$ ) you use the transformation relationships between the two sets of coordinates:
$x = r \cdot cos (θ)$ $x=r*cos(theta)$
$y = r \cdot sin (θ)$ $y=r*sin(theta)$
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You have to remember that your equation may need some algebraic/trigonometric manipulations before being transformed into rectangular form; for example, consider:

$r [- 2 sin (θ) + 3 cos (θ)] = 2$ $r[-2sin(theta)+3cos(theta)]=2$
$- 2 r sin (θ) + 3 r cos (θ) = 2$ $-2rsin(theta)+3rcos(theta)=2$

Now you use the above transformations, and get:

$- 2 y + 3 x = 2$ $-2y+3x=2$
Which is the equation of a straight line!

A more complicated situation can be the following example:
$θ + \frac{π}{4} = 0$ $theta+pi/4=0$
You can write:
$θ = - \frac{π}{4}$ $theta=-pi/4$
Take the tangent of both sides and multiply and divide by $r$ $r$ the left side:
$\frac{r}{r} \cdot tan (θ) = tan (- \frac{π}{4})$ $r/r*tan(theta)=tan(-pi/4)$
$\frac{r sin (θ)}{r cos (θ)} = - 1$ $(rsin(theta))/(rcos(theta))=-1$
Transforming you get:
$\frac{y}{x} = - 1$ $y/x=-1$
$y = - x$ $y=-x$

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How do you convert polar equations to rectangular equations?

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