Question

Write the equation in rectangular coordinates: `r=10sin(θ)`?

Answer 1

`x^2+(y-5)^2= 5^2`

Explanation:

Given: `r=10sin(theta)`

This is the graph in polar coordinates:

![www.desmos.com/calculator]()

Multiply both sides by `r`:

`r^2=10rsin(theta)`

Substitute `r^2 = x^2+y^2` and `rsin(theta) = y`

`x^2+y^2=10y`

Add `-10y+ k^2` to both sides:

`x^2+y^2-10y+k^2=k^2" [1]"`

Matching the right side of the pattern `(y-k)^2=y^2-2ky+k^2` with `y^2-10y+k^2`

We observe that the following equation will allow us to determine the value of k:

`-2ky = -10y`

`k = 5`

Substitute the left side of the pattern into equation [1] with `k = 5`:

`x^2+(y-5)^2= 5^2" [1.1]"`

This is the Cartesian equation of a circle with center `(0,5)` and radius, `r =5`

This is the graph in Cartesian coordinates:

![www.desmos.com/calculator]()

Please observe that the graphs are identical, therefore, the conversion is correct.

Answer 2

`x^2+(y-5)^2=25`

Explanation:

`"to convert from "color(blue)"polar to rectangular"`

`•color(white)(x)r=sqrt(x^2+y^2)rArrr^2=x^2+y^2`

`•color(white)(x)y=rsinthetarArrsintheta=y/r`

`r=10sintheta`

`rArrr=(10y)/r`

`"multiply both sides by r"`

`rArrr^2=10y`

`rArrx^2+y^2-10y=0`

`"completing the square on "y^2-10y" gives"`

`rArrx^2+(y-5)^2=25`

`"which is the equation of a circle"`

`"centre "=(0,5)" and radius "=5`
graph{x^2+(y-5)^2=25 [-20, 20, -10, 10]}