2022-02-01

How do you write the trigonometric form of $\frac{5}{2}\left(\sqrt{3}-i\right)$?

Troy Sutton

Expert

Answer: $5\left(\mathrm{cos}\left(-\frac{\pi }{6}\right)+i\mathrm{sin}\left(-\frac{\pi }{6}\right)\right)$
Explanation:
The trigonometric form of a complex number $z=a+ib$ is
$z=|z|\left(\mathrm{cos}\theta +i\mathrm{sin}\theta \right)$
where $\mathrm{cos}\theta =\frac{a}{|z|}$
and $\mathrm{sin}\theta =\frac{b}{|z|}$
Here $z=\frac{5}{2}\left(\sqrt{3}-i\right)$
$|z|=\frac{5}{2}\sqrt{3+1}=\frac{5}{2}\cdot 2=5$
$z=5\left(\frac{\sqrt{3}}{2}-\frac{1}{2}i\right)$
$\mathrm{cos}\theta =\frac{\sqrt{3}}{2}$
$\mathrm{sin}\theta =-\frac{1}{2}$
$\theta =-\frac{\pi }{6},\left[\text{mod}2\pi \right]$
The trigonometric form is
$z=5\left(\mathrm{cos}\left(-\frac{\pi }{6}\right)+i\mathrm{sin}\left(-\frac{\pi }{6}\right)\right)=5{e}^{-i\frac{\pi }{6}}$

Do you have a similar question?