Question

How do you convert $\left( 4,-2 \right)$ from Cartesian to polar coordinates?

Answer 1

In this question we have to convert the given Cartesian coordinates to the polar coordinates. In order to convert first of all we will find $r$ as the distance between origin and the Cartesian point as $r=\sqrt{{{x}^{2}}+{{y}^{2}}}$ where $\left( x,y \right)$ are Cartesian coordinates and $\theta$ is the angle the ray joining the origin and the point makes with positive x-axis.

Complete step-by-step answer:
We have been given a Cartesian coordinates $\left( 4,-2 \right)$.
We have to convert the given Cartesian coordinates to polar coordinates.
We know that we can convert the Cartesian coordinates $\left( x,y \right)$ to polar coordinates $\left( r,\theta \right)$ of a point using the following relation
$r=\sqrt{{{x}^{2}}+{{y}^{2}}}$
$\theta =atan2\left( y,x \right)$
Here the function $atan2\left( y,x \right)$ called 2-argument inverse tangent and is defined as
\theta =\left\\{ \begin{aligned} & {{\tan }^{-1}}\left( \dfrac{y}{x} \right)\text{ if }x>0 \\\ & {{\tan }^{-1}}\left( \dfrac{y}{x} \right)\text{+}\pi \text{ if }x<0\text{ and y}\ge 0 \\\ & {{\tan }^{-1}}\left( \dfrac{y}{x} \right)-\pi \text{ if }x<0\text{ and y}<0 \\\ & \dfrac{\pi }{2}\text{ if }x=0\text{ and y}>0 \\\ & -\dfrac{\pi }{2}\text{ if }x=0\text{ and y}>0 \\\ & undefined\text{ if }x=0\text{ and y=}0 \\\ \end{aligned} \right\\}
We have given the Cartesian coordinates $\left( x,y \right)=\left( 4,-2 \right)$.
Now, we can find the value of r by using the relation $r=\sqrt{{{x}^{2}}+{{y}^{2}}}$
Now, substituting the values we will get
$\Rightarrow r=\sqrt{{{4}^{2}}+{{\left( -2 \right)}^{2}}}$
Simplifying the above obtained equation we will get
$\begin{aligned} & \Rightarrow r=\sqrt{16+4} \\\ & \Rightarrow r=\sqrt{20} \\\ & \Rightarrow r=2\sqrt{5} \\\ \end{aligned}$
Now, we can find the value of $\theta$ as $\theta ={{\tan }^{-1}}\left( \dfrac{-2}{4} \right)$
Simplifying the above obtained equation we will get
$\begin{aligned} & \Rightarrow \theta ={{\tan }^{-1}}\left( \dfrac{-1}{2} \right) \\\ & \Rightarrow \theta =-{{\tan }^{-1}}\left( \dfrac{1}{2} \right) \\\ & \Rightarrow \theta =-0.463 \\\ \end{aligned}$
So the polar coordinates are$\left( 2\sqrt{5},-0.463 \right)$.
Hence we get the required polar coordinates as $\left( 2\sqrt{5},-0.463 \right)$.

Note: The point to be noted is that r also called as radical coordinate is always positive and the value of $\theta$ also called angular coordinate is always measured in radians. We can also convert the negative angle to positive by adding $2\pi$.