<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Skip Navigation
Our Terms of Use (click here to view) have changed. By continuing to use this site, you are agreeing to our new Terms of Use.

Equations Using DeMoivre's Theorem

Complex roots of an equation.

Atoms Practice
Estimated10 minsto complete
Practice Equations Using DeMoivre's Theorem
Estimated10 minsto complete
Practice Now
Turn In
Polar Theorems

Feel free to modify and personalize this study guide by clicking “Customize.”

Theorem Overview

In your own words, describe each theorem as it relates to polar equations.
Theorem Description
Product Theorem ________________________________________________________________
Quotient Theorem ________________________________________________________________
DeMoivre's Theorem ________________________________________________________________


Complete the theorems:

Product Theorem

\begin{align*}r_1(\cos \theta_1 + i \sin \theta_1) \cdot r_2(\cos \theta_2 + i \sin \theta_2)=\end{align*} ____________________________________


Quotient Theorem

\begin{align*}\frac{r_1(\cos \theta_1+i \sin \theta_1)}{r_2(\cos \theta_2+i \sin \theta_2)}=\end{align*} ____________________________________


DeMoivre's Theorem

\begin{align*}z^n = [r(\cos \theta + i \sin \theta)]^n =\end{align*} ____________________________________

Where \begin{align*}z = r(\cos \theta + i \sin \theta)\end{align*} and let \begin{align*}n\end{align*} be a positive integer.


 The general rule for finding the \begin{align*}n^{th}\end{align*} roots of a complex number if \begin{align*}z = r(\cos \theta + i \sin \theta)\end{align*} is _____________________________________________ (Hint: start wth DeMoivre's Theorem).


Multiply each pair of complex numbers. If they are not in trigonometric form, change them before multiplying.

  1. \begin{align*}-3(\cos 70^\circ+i\sin 70^\circ)\cdot 3(\cos 85^\circ +i\sin 85^\circ )\end{align*}
  2. \begin{align*}7(\cos 85^\circ+i\sin 85^\circ)\cdot \sqrt{2}(\cos 40^\circ +i\sin 40^\circ )\end{align*}
  3. \begin{align*}(3-2i)\cdot (1+i)\end{align*}

Divide each pair of complex numbers. If they are not in trigonometric form, change them before dividing.
  1. \begin{align*}\frac{-3(\cos 70^\circ+i\sin 70^\circ)}{3(\cos 85^\circ +i\sin 85^\circ )}\end{align*}
  2. \begin{align*}\frac{7(\cos 85^\circ+i\sin 85^\circ)}{\sqrt{2}(\cos 40^\circ +i\sin 40^\circ )}\end{align*}
  3. \begin{align*}\frac{(3-2i)}{(1+i)}\end{align*}
Use DeMoivre's Theorem to evaluate each expression. Write your answer in standard form.
  1. \begin{align*}[3(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4})]^5\end{align*}
  2. \begin{align*}(2-\sqrt{5}i)^5\end{align*}
  3. \begin{align*}(\sqrt{2}+\sqrt{2}i)^4\end{align*}

Find the principal fifth roots of each complex number. Write your answers in standard form.

  1. \begin{align*}32(\cos \frac{\pi}{4}+i\sin \frac{\pi}{4})\end{align*}
  2. \begin{align*}2(\cos \frac{\pi}{3}+i\sin \frac{\pi}{3})\end{align*}
  3. \begin{align*}32i\end{align*}
Solve each equation.
  1. \begin{align*}x^3=343\end{align*}
  2. \begin{align*}x^7=-128\end{align*}
  3. \begin{align*}x^4+5=86\end{align*}

Click here for more help with equations using DeMoivre's Theorem.

Explore More

Sign in to explore more, including practice questions and solutions for Product Theorem.
Please wait...
Please wait...