Answer:
[tex]x^2+(y-1)^2=1[/tex]
Step-by-step explanation:
The given cylindrical equation is
[tex]r=2\sin \theta[/tex]
Multiply both sides by r.
[tex]r^2=2r\sin \theta[/tex] .... (1)
The required formulas are
[tex]x=r\cos \theta,y=r\sin \theta,r^2=x^2+y^2[/tex]
Substitute [tex]r\sin \theta =y,r^2=x^2+y^2[/tex] in equation (1).
[tex]x^2+y^2=2y[/tex]
[tex]x^2+y^2-2y=0[/tex]
Add 1 on both sides.
[tex]x^2+(y^2-2y+1)=1[/tex]
[tex](x-0)^2+(y-1)^2=1^2[/tex] [tex][\because (a-b)^2=a^2-2ab+b^2][/tex]
It is the equation of a circle centered at (0,1) with radius 1.
Therefore, the equation in rectangular coordinates is [tex]x^2+(y-1)^2=1[/tex].
