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Exercises Trigonometry

Example 11 from Algebra and Trigonometry, 10.5 Polar Form of Complex Numbers

Finding the nth Root of a Complex Number

Evaluate the cube roots of $z = 8 (\cos (\frac{2 π}{3}) + i \sin (\frac{2 π}{3})) .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We have

$\begin{array}{l} z^{\frac{1}{3}} = 8^{\frac{1}{3}} [\cos (\frac{\frac{2 π}{3}}{3} + \frac{2 k π}{3}) + i \sin (\frac{\frac{2 π}{3}}{3} + \frac{2 k π}{3})] \\ z^{\frac{1}{3}} = 2 [\cos (\frac{2 π}{9} + \frac{2 k π}{3}) + i \sin (\frac{2 π}{9} + \frac{2 k π}{3})] \end{array}$
There will be three roots: $k = 0, 1, 2.$ When $k = 0,$ we have

$z^{\frac{1}{3}} = 2 (\cos (\frac{2 π}{9}) + i \sin (\frac{2 π}{9}))$
When $k = 1,$ we have

$\begin{array}{l} z^{\frac{1}{3}} = 2 [\cos (\frac{2 π}{9} + \frac{6 π}{9}) + i \sin (\frac{2 π}{9} + \frac{6 π}{9})] \begin{matrix} \end{matrix} Add \frac{2 (1) π}{3} to each angle. \\ z^{\frac{1}{3}} = 2 (\cos (\frac{8 π}{9}) + i \sin (\frac{8 π}{9})) \end{array}$
When $k = 2,$ we have
$\begin{array}{l} z^{\frac{1}{3}} = 2 [\cos (\frac{2 π}{9} + \frac{12 π}{9}) + i \sin (\frac{2 π}{9} + \frac{12 π}{9})] \begin{matrix} \end{matrix} & Add \frac{2 (2) π}{3} to each angle. \\ z^{\frac{1}{3}} = 2 (\cos (\frac{14 π}{9}) + i \sin (\frac{14 π}{9})) \end{array}$
Remember to find the common denominator to simplify fractions in situations like this one. For $k = 1,$ the angle simplification is

$\begin{array}{l} \frac{\frac{2 π}{3}}{3} + \frac{2 (1) π}{3} = \frac{2 π}{3} (\frac{1}{3}) + \frac{2 (1) π}{3} (\frac{3}{3}) \\ = \frac{2 π}{9} + \frac{6 π}{9} \\ = \frac{8 π}{9} \end{array}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Find the four fourth roots of $16 (\cos (120°) + i \sin (120°)) .$

How did it go?

PreviousEvaluating an Expression Using De Moivre’s Theorem Next Parameterizing a Curve

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 10.5, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.