weird conversion of polar coordinates into rectangular coordinates.

Question

Given $r=\frac{4}{1-\cos(\theta)}$, convert into rectangular (cartesian) coordinates.

My solution:

1. Square both sides: $r^2=\frac{16}{\sin^2(\theta)}$

2. Multiply the denominator by $r^2$ to get: $r^2\sin^2(\theta)=16$

3. Since $y=r\sin(\theta)$, we can make $r^2\sin^2(\theta)=y^2$, thus $y^2=16$ or $y^2-16=0$

But the answer in the book does not match mine, can you explain to me what I am doing wrong?

Answer 1

Your mistake is in step 1, squaring both sides: note that $$(1-\cos(\theta))^2\neq 1 - \cos^2(\theta).$$ The correct result of squaring both sides would be $$r^2=\frac{16}{(1-\cos(\theta))^2}=\frac{16}{1-2\cos(\theta)+\cos^2(\theta)}$$