Question

Convert $(2,2\sqrt{3})$ from rectangular coordinates to polar coordinates $(r,\theta )$, where $r>0$ and $0\le \theta <2\pi$.

Short answer

The polar point is $\left(4,\frac{\pi }{3}\right)$.

Step-by-step solution

Step 1. Given information 

We are given a point $\left(2,2\sqrt{3}\right)$.

We have to convert the given point from rectangular coordinates to polar coordinates.

Step 2. Converting the point

We know that,

$r^2=x^2+y^2$

So, putting values, we get

$$\begin{gathered} r^2=2^2+(2\sqrt{3}{)}^2 \\ r=\sqrt{4+12} \\ r=\sqrt{16} \\ r=4 \end{gathered}$$

Step 3. Converting the point

Also, we know that

$\tan \theta =\frac{y}{x}$

so, putting values, we get

$$\begin{gathered} \tan \theta =\frac{2\sqrt{3}}{2} \\ \theta ={\tan }^{-1}\sqrt{3} \\ \theta =\frac{\pi }{3} \end{gathered}$$

Hence, the polar point is role="math" localid="1647002989966" $\left(4,\frac{\pi }{3}\right)$.