We need to sketch `r=sin theta-1`.

Using the same process for the earlier examples, we obtain:

30°60°90°120°150°180°210°240°270°300°330°0123

Graph of r = sin θ − 1, a limacon.

For the curve above, when θ = 0, `r = −1`, so the curve starts on the left side of the origin.

Conversion to Rectangular Form

To convert to rectangular form, we use r2 = x2 + y2 and

`sin^2theta=(y^2)/(r^2)=(y^2)/(x^2+y^2)`

In rectangular form, r = sin θ − 1 is:

`sqrt(x^2+y^2)=y/sqrt(x^2+y^2)-1`

`x^2+y^2=y-sqrt(x^2+y^2)`

`x^2+y^2-y=-sqrt(x^2+y^2)`

`(x^2+y^2-y)^2=x^2+y^2`

`x^4+2x^2y^2+y^4-2y(x^2+y^2)+y^2` `=x^2+y^2`

`x^4+2x^2y^2+y^4-2y(x^2+y^2)-x^2` `=0`