The letters r and theta represent polar coordinates. Write each equation using rectangular coordinates (x,y).
(1) r = sin(theta) + 1
(2) r = sin(theta) - cos(theta)
(2) Substitute $\displaystyle x = r \cos \theta \Rightarrow \cos \theta = \frac{x}{r}$ and $\displaystyle y = r \sin \theta \Rightarrow \sin \theta = \frac{y}{r}$.
Multiply both sides by r.
Substitute $\displaystyle r^2 = x^2 + y^2$.
(1) Substitute $\displaystyle y = r \sin \theta \Rightarrow \sin \theta = \frac{y}{r}$.
Multiply both sides by r.
Substitute $\displaystyle r = \sqrt{x^2 + y^2} \Rightarrow r^2 = x^2 + y^2$.