Answer
The complex numbers in rectangular form $2+2i$, $1+\sqrt{3}i$, $2-2i$, $2\sqrt{3}-2i$ are written in polar form as $2\left( \cos \frac{\pi }{3}+i\sin \frac{\pi }{3} \right)$, $2.82\left( \cos \frac{7\pi }{4}+i\sin \frac{7\pi }{4} \right)$ and $4\left( \cos \frac{11\pi }{6}+i\sin \frac{11\pi }{6} \right)$ respectively.
Work Step by Step
Any complex number $\text{a}+i\text{b}$ can be written in the polar form $r{{e}^{i\theta }}$ using a graphing utility calculator.
Consider the complex number $2+2i$. Use the graphing utility calculator to obtain the results.
Go to the main screen of the calculator and enter the complex number $2+2i$. Press $\left[ \text{MATH} \right]$ and then the right arrow twice and then press $\left[ 7 \right]$. Then press $\left[ \text{ENTER} \right]$ to get the polar form of the number. It gives $2.82{{e}^{0.785i}}$ or it can be rewritten as $2.82\left( \cos \frac{\pi }{4}+i\sin \frac{\pi }{4} \right)$.
Now, consider the complex number $1+\sqrt{3}i$. Use the graphing utility calculator to obtain the results.
Go to the main screen of the calculator and enter the complex number $1+\sqrt{3}i$. Press $\left[ \text{MATH} \right]$ and then the right arrow twice and then press $\left[ 7 \right]$. Then press $\left[ \text{ENTER} \right]$ to get the polar form of the number. It gives $2{{e}^{1.047i}}$ or it can be rewritten as $2\left( \cos \frac{\pi }{3}+i\sin \frac{\pi }{3} \right)$.
Now, consider the complex number $2-2i$. Use the graphing utility calculator to obtain the results.
Go to the main screen of the calculator and enter the complex number $2-2i$. Press $\left[ \text{MATH} \right]$ and then the right arrow twice and then press $\left[ 7 \right]$. Then press $\left[ \text{ENTER} \right]$ to get the polar form of the number. It gives $2.82{{e}^{-0.785i}}$ . Since the angle $\theta $ lies in the fourth quadrant, $\theta =2\pi -\frac{\pi }{4}=\frac{7\pi }{4}$ or it can be rewritten as $2.82\left( \cos \frac{7\pi }{4}+i\sin \frac{7\pi }{4} \right)$
Now, consider the complex number $2\sqrt{3}-2i$. Use the graphing utility calculator to obtain the results.
Go to the main screen of the calculator and enter the complex number $2\sqrt{3}-2i$. Press $\left[ \text{MATH} \right]$ and then the right arrow twice and then press $\left[ 7 \right]$. Then press $\left[ \text{ENTER} \right]$ to get the polar form of the number. It gives $4{{e}^{-0.523i}}$ . Since the angle $\theta $ lies in the fourth quadrant, $\theta =2\pi -\frac{\pi }{6}=\frac{11\pi }{6}$ or it can be rewritten as $4\left( \cos \frac{11\pi }{6}+i\sin \frac{11\pi }{6} \right)$.
