Question

In Problems55–66, the rectangular coordinates of a point are given. Find polar coordinates for each point.

$\left(0,-2\right)$

Short answer

The polar coordinates is $\left(2,-\frac{\pi }{2}\right)$.

Step-by-step solution

Step 1. Given Information  

We are given rectangular coordinates of a point,

$\left(0,-2\right)$

We have to find polar coordinates of given point.

Step 2. Finding the polar coordinates  

The point $(0,-2)$lies in the third quadrant, so

$$\begin{gathered} x=r\cos \theta \\ r\cos \theta =0-\left(1\right) \\ y=r\sin \theta \\ r\sin \theta =-2-\left(2\right) \end{gathered}$$

Squaring both side and adding the two equations, we get

$$\begin{gathered} (r\cos \theta {)}^2+(r\sin \theta {)}^2=0^2+(-2{)}^2 \\ {r}^2\left({\cos }^2\theta +{\sin }^2\theta \right)=4 \\ {r}^2=4 \\ r=2 \end{gathered}$$

Step 3. Finding the polar coordinates 

We know that,

$$\begin{gathered} \tan \theta =\frac{y}{x}\text{, so} \\ \theta ={\tan }^{-1}\frac{y}{x},-\frac{\pi }{2}<\theta <\frac{\pi }{2} \end{gathered}$$

Putting values, we get

role="math" localid="1646730385944" $$\begin{gathered} \theta ={\tan }^{-1}\frac{-2}{0} \\ \theta ={\tan }^{-1}-\infty \\ \theta =-\frac{\pi }{2} \end{gathered}$$

So, the polar coordinate will be $\left(2,-\frac{\pi }{2}\right)$.