Question

Polar Equations to Rectangular Equations

Convert the polar equation to rectangular coordinates

$r=4\sin \theta$

Short answer

The rectangular equation is$x^2+{(y-2)}^2=4$

Step-by-step solution

Step 1. Given information

An equation is given in polar coordinate system as$r=4\sin \theta$

Step 2. Concept used

Use conversion formulae from rectangular coordinates into polar coordinates as-

$\begin{array}{l}r=\sqrt{x^2+y^2}\\ \theta ={\tan }^{-1}\left(\frac{y}{x}\right)\end{array}$

Substitute the value in the formulae. And simplify them.

Step 3. Calculation

The polar equation is given as,$r=4\sin \theta$

Multiply by r on both sides,

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

Formulae used,

$\begin{array}{l}{r}^2=x^2+y^2\\ y=r\sin \theta \end{array}$

Substitute the values in the formulae,

$\begin{array}{l}\Rightarrow x^2+y^2=4y\\ x^2+y^2-4y=0\\ \Rightarrow x^2+(y^2-4y+4)=4\\ x^2+({\left(y\right)}^2-2\times 2\times y+{\left(2\right)}^2)=4\\ x^2\text{Invalid <m:msup> element}+=4\end{array}$

It is a circle with radius of two units