The relation between cartesian coordinates (x, y) and polar coordinates (r, θ), is given by the following equations:
[tex]\begin{gathered} x=r\cdot\cos \theta, \\ y=r\cdot\sin \theta, \\ r^2=x^2+y^2. \end{gathered}[/tex]
We must convert the following equation from polar coordinates to cartesian coordinates:
[tex]r=-4\cdot\sin \theta.[/tex]
1) We multiply both sides of the equation by r:
[tex]\begin{gathered} r\cdot r=r\cdot(-4\cdot\sin \theta)\text{.} \\ r^2=-4\cdot(r\cdot\sin \theta). \end{gathered}[/tex]
2) Now, we replace by the identities above:
[tex]x^2+y^2=-4\cdot y.[/tex]
3) We rewrite the equation in the following way:
[tex]\begin{gathered} x^2+y^2+4y=0, \\ x^2+(y^2+2\cdot2y+4)-4=0, \\ x^2+(y+2)^2-4=0, \\ x^2+(y+2)^2=4. \end{gathered}[/tex]
Answer
C.
[tex]x^2+(y+2)^2=4[/tex]
