Question

16) Express the complex number in trigonometric form.

-3i

A) 3(cos 180° + i sin 180°)
B) 3(cos 270° + i sin 270°)
C) 3(cos 90° + i sin 90°)
D) 3(cos 0° + i sin 0°)

Answer 1

$\bf -3i\implies 0-3i\implies \stackrel{a}{0}~~~~\stackrel{b}{-3}~i\quad \begin{cases} r=\sqrt{a^2+b^2}\\\\ \theta =tan^{-1}\left( \frac{b}{a} \right) \end{cases} \\\\\\ r=\sqrt{0^2+(-3)^2}\implies r=\sqrt{9}\implies r=3$

now, as far as the angle θ, if we plug those values, we'd get an undefined, it just so happen that tan⁻¹ is not defined on that range, however, let's just use the provided coordinates, check the picture below.

therefore   $\bf z=r[cos(\theta )+i~sin(\theta )]\implies z=3[cos(270^o)+i~sin(270^o)]$