How do you convert polar equations to rectangular equations?

1 Answer
Gió
Jan 8, 2015

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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