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

Let's find the answer using the following interactive graph. You will trace out the required curve as you change the angle.

**Drag** the blue dot (slowly) left and right to change the angle *θ* and observe the resulting changes in the value of *r*.

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**Hint:** Let go of the blue dot to see a smooth curve at any time. You can trace out positive or negative angles.

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

In case you can't see the graph above, here's a static version of it:

To convert to rectangular form, we use *r*^{2} = *x*^{2} + *y*^{2} 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`

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