Question

Find an equation in rectangular coordinates for the cylindrical equation r = 2sintheta

Answer 1

Answer:

$x^2+(y-1)^2=1$

Step-by-step explanation:

The given cylindrical equation is

$r=2\sin \theta$

Multiply both sides by r.

$r^2=2r\sin \theta$           .... (1)

The required formulas are

$x=r\cos \theta,y=r\sin \theta,r^2=x^2+y^2$

Substitute $r\sin \theta =y,r^2=x^2+y^2$ in equation (1).

$x^2+y^2=2y$

$x^2+y^2-2y=0$

Add 1 on both sides.

$x^2+(y^2-2y+1)=1$

$(x-0)^2+(y-1)^2=1^2$                    $[\because (a-b)^2=a^2-2ab+b^2]$

It is the equation of a circle centered at (0,1) with radius 1.

Therefore, the equation in rectangular coordinates is $x^2+(y-1)^2=1$.