Question

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

$\left(1,-1\right)$

Short answer

The polar coordinates is $\left(\sqrt{2},-\frac{\pi }{4}\right)$.

Step-by-step solution

Step 1. Given Information   

We are given rectangular coordinates of a point,

$\left(1,-1\right)$

We have to find polar coordinates of given point.

Step 2. Finding the polar coordinates   

The distance of the point $\left(1,-1\right)$from origin is given by,

$$\begin{gathered} r=\sqrt{x^2+y^2} \\ r=\sqrt{1^2+(-1{)}^2} \\ =\sqrt{1+1} \\ =\sqrt{2} \end{gathered}$$

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

$$\begin{gathered} \theta ={\tan }^{-1}\left(\frac{-1}{1}\right) \\ ={\tan }^{-1}(-1) \\ =-\frac{\pi }{4} \end{gathered}$$

So, the polar coordinate will be$\left(\sqrt{2},-\frac{\pi }{4}\right)$.