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Exercises Trigonometry

Example 5 from Algebra and Trigonometry, 10.5 Polar Form of Complex Numbers

Finding the Polar Form of a Complex Number

Find the polar form of $- 4 + 4 i .$

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
First, find the value of $r .$

$\begin{array}{l} r = \sqrt{x^{2} + y^{2}} \\ r = \sqrt{{(- 4)}^{2} + (4^{2})} \\ r = \sqrt{32} \\ r = 4 \sqrt{2} \end{array}$
Find the angle $θ$ using the formula:
$\begin{array}{l} \cos θ = \frac{x}{r} \\ \cos θ = \frac{- 4}{4 \sqrt{2}} \\ \cos θ = - \frac{1}{\sqrt{2}} \\ θ = \cos^{- 1} (- \frac{1}{\sqrt{2}}) = \frac{3 π}{4} \end{array}$
Thus, the solution is $4 \sqrt{2} cis (\frac{3 π}{4}) .$

Your turn

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

Write $z = \sqrt{3} + i$ in polar form.

How did it go?

PreviousExpressing a Complex Number Using Polar Coordinates Next Converting from Polar to Rectangular Form

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.