Question

How do you convert $r=4\sin \theta$ to rectangular form?

Answer 1

Convert polar form to rectangular form:

The given polar form is $r=4\sin \theta$.

Multiply each side by $r$.

$r^2=4r\sin \theta$

Substitute $y$ for $r\sin \theta$ in $r^2=4r\sin \theta$:

$r^2=4y$

The equation of circle is $x^2+y^2=r^2$ whose center is at the origin.

Substitute $4y$ for $r^2$ in $x^2+y^2=r^2$:

$\begin{array}{rcl}{x}^2+y^2& =& 4y\\ & \Rightarrow & x^2+y^2-4y+4=4\text{[Subtracting 4y and Adding 4 on both side]}\\ & \Rightarrow & x^2+{\left(y-2\right)}^2=4\left[{\left(a-b\right)}^2=a^2-2ab+b^2\right]\\ & \Rightarrow & x^2+{\left(y-2\right)}^2=2^2\end{array}$

The above equation has center $\left(0,2\right)$ and radius $2$.

Hence, the rectangular form of the given polar from is $\begin{array}{rcl}{\mathit{x}}^{\mathbf{2}}\mathbf{+}{\left(y-2\right)}^{\mathbf{2}}& \mathbf{=}& \mathbf{4}\end{array}$.