Question

How do you convert polar equations to rectangular equations?

Answer 1

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)`

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`