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Exercises Trigonometry

Example 4 from Algebra and Trigonometry, 10.5 Polar Form of Complex Numbers

Expressing a Complex Number Using Polar Coordinates

Express the complex number $4 i$ using polar coordinates.

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
On the complex plane, the number $z = 4 i$ is the same as $z = 0 + 4 i .$ Writing it in polar form, we have to calculate $r$ first.

$\begin{array}{l} r = \sqrt{x^{2} + y^{2}} \\ r = \sqrt{0^{2} + 4^{2}} \\ r = \sqrt{16} \\ r = 4 \end{array}$
Next, we look at $x .$ If $x = r \cos θ,$ and $x = 0,$ then $θ = \frac{π}{2} .$ In polar coordinates, the complex number $z = 3 i$ can be written as $z = 4 (\cos (\frac{π}{2}) + i \sin (\frac{π}{2}))$ or $4 cis (\frac{π}{2}) .$ See Figure 6.

Figure 6

Your turn

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

Express $z = 3 i$ as $r cis θ$ in polar form.

How did it go?

PreviousFinding the Absolute Value of a Complex Number Next Finding the Polar Form of a Complex Number

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.