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.

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:

### 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