Question

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

$\left(-0.8,-2.1\right)$

Short answer

The polar coordinates is $\left(\sqrt{5.05},4.34\right)$.

Step-by-step solution

Step 1. Given Information

We are given rectangular coordinates of a point,

$\left(-0.8,-2.1\right)$

We have to find polar coordinates of given point.

Step 2. Finding the polar coordinates    

The distance of the point $\left(-0.8,-2.1\right)$from origin is given by,

$$\begin{gathered} r=\sqrt{x^2+y^2} \\ r=\sqrt{{(-0.8)}^2+{\left(-2.1\right)}^2} \\ =\sqrt{0.64+4.41} \\ =\sqrt{5.05} \end{gathered}$$

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

localid="1646797452204" $$\begin{gathered} \theta ={\tan }^{-1}\left(\frac{-2.1}{-0.8}\right) \\ ={\tan }^{-1}(2.625) \\ =-1.2 \end{gathered}$$

Since, the point is in the third quadrant, so by adding $\pi =3.14$, we get the angle in third quadrant.

$$\begin{gathered} \theta =3.14+1.2 \\ \theta =4.34 \end{gathered}$$

So, the polar coordinate will be role="math" localid="1646797577326" $\left(\sqrt{5.05},4.34\right)$.