Question

How do you find the cartesian graph of `r = 5sin(θ)`?

Answer 1

You basically get a circle.

Explanation:

Consider the following diagram:

We can see that the relationships between rectangular and polar coordinates are:
`r=sqrt(x^2+y^2)`
`theta=arctan(y/x)`
and:
`x=rcos(theta)`
`y=rsin(theta)`

Given our expression:
`r=5sin(theta)`

multiply by `r` both sides:
`r^2=5rsin(theta)`
so that you get, using our relationships of conversion:

`color(red)(x^2+y^2=5y)` (1)

which is the equation of a circle centered at `(x_c=0,y_c=5/2)` and with radius `r=5/2`, whose equation is found from the general form of a circle:
`(x-x_c)^2+(y-y_c)^2=r^2`
(try to substitute the values of the center and radius and you'll find (1)).
You can now plot it directly.

Next, to have some fun, I used Excel to evaluate, using our relationships of conversion, the coordinates `x` and `y` (in red) and plot them: