Question

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

Answer 1

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

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`.