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 $r$ and $theta$ ) into rectangular form (in $x$ and $y$ ) you use the transformation relationships between the two sets of coordinates:
$x=r*cos(theta)$
$y=r*sin(theta)$
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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:

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

Now you use the above transformations, and get:

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

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

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How do you convert polar equations to rectangular equations?

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