Answer
The complex numbers in polar form $6\left( \cos 30{}^\circ +i\sin 30{}^\circ \right)$, $12\left( \cos 60{}^\circ +i\sin 60{}^\circ \right)$, $4\left( \cos 240{}^\circ +i\sin 240{}^\circ \right)$, $10\left( \cos 210{}^\circ +i\sin 210{}^\circ \right)$ are written in rectangular form as $5.19+3i$, $6+10.39i$, $-2-3.46i$ and $-8.66-5i$ respectively
Any complex number $r\left( \cos \theta +i\sin \theta \right)$ can be written in the rectangular form $\text{a}+i\text{b}$ using a graphing utility calculator.
Work Step by Step
Consider the complex number $6\left( \cos 30{}^\circ +i\sin 30{}^\circ \right)$. Use the graphing utility calculator to obtain the results.
Go to the main screen of the calculator and enter the complex number $6\left( \cos 30{}^\circ +i\sin 30{}^\circ \right)$. Press $\left[ \text{MATH} \right]$ and then the right arrow twice and then press $\left[ 6 \right]$. Then press $\left[ \text{ENTER} \right]$ to get the polar form of the number. It gives $5.19+3i$.
Now, consider the complex number $12\left( \cos 60{}^\circ +i\sin 60{}^\circ \right)$. Use the graphing utility calculator to obtain the results.
Go to the main screen of the calculator and enter the complex number $12\left( \cos 60{}^\circ +i\sin 60{}^\circ \right)$. Press $\left[ \text{MATH} \right]$ and then the right arrow twice and then press $\left[ 6 \right]$. Then press $\left[ \text{ENTER} \right]$ to get the polar form of the number. It gives $6+10.39i$.
Now, consider the complex number $4\left( \cos 240{}^\circ +i\sin 240{}^\circ \right)$. Use the graphing utility calculator to obtain the results.
Go to the main screen of the calculator and enter the complex number $4\left( \cos 240{}^\circ +i\sin 240{}^\circ \right)$. Press $\left[ \text{MATH} \right]$ and then the right arrow twice and then press $\left[ 6 \right]$. Then press $\left[ \text{ENTER} \right]$ to get the polar form of the number. It gives $-2-3.46i$.
Now, consider the complex number $10\left( \cos 210{}^\circ +i\sin 210{}^\circ \right)$. Use the graphing utility calculator to obtain the results.
Go to the main screen of the calculator and enter the complex number $10\left( \cos 210{}^\circ +i\sin 210{}^\circ \right)$. Press $\left[ \text{MATH} \right]$ and then the right arrow twice and then press $\left[ 6 \right]$. Then press $\left[ \text{ENTER} \right]$ to get the polar form of the number. It gives $-8.66-5i$.
