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Equations Using DeMoivre's Theorem

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Polar Theorems

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Theorem Overview

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

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Complete the theorems:

Product Theorem

r_1(\cos \theta_1 + i \sin \theta_1) \cdot r_2(\cos \theta_2 + i \sin \theta_2)= ____________________________________

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Quotient Theorem

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

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DeMoivre's Theorem

z^n = [r(\cos \theta + i \sin \theta)]^n = ____________________________________

Where z = r(\cos \theta + i \sin \theta) and let n be a positive integer.

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


Practice

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

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

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

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

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

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

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