There are no values given for $r$, or $\theta$. How do I derive the cartesian equation for this? It's a question from a textbook I have.
2026-03-30 04:25:20.1774844720
What is the cartesian equation of $r = 1 + r \sin(\theta)?$
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Take the polar function: $$r = 1 + r \sin \theta$$
Square everything: $$r^2 = 1 + 2 r \sin\theta + r^2 \sin^2 \theta$$
Substitute $(r^2 = x^2+y^2)$ and $(r \sin\theta = y)$: $$x^2 + y^2 = 1 + 2 y + y^2$$
Cancel common terms: $$x^2 = 1 + 2y$$
Rearrange into an expression of y as a function of x: $$y = \frac{x^2-1} 2$$