Question

Express the complex number in trigonometric form.

-2i

Answer 1

given z=a+ib u have to compute
\[\large \rho=|z|=\sqrt{a^2+b^2} \]
and
\[\large \varphi=\text{Arg}(z) \]
which is the angle between the positive side of the x-axis and the vector z

Answer 2

then
\[\large z=\rho(\cos\varphi+i\sin\varphi) \]

Answer 3

\(\bf -2i\implies 0-2i\qquad \textit{trigonometric form}\implies \color{red}{r[cos(\theta)+isin(\theta)]}\\ \quad \\
\begin{array}{ccllll}
a&b\\
0&-2i
\end{array}\\ \quad \\
\implies r =\sqrt{a^2+b^2}\\ \quad \\
tan(\theta)=\cfrac{b}{a}\implies tan^{-1}[tan(\theta)]=tan^{-1}\left(\cfrac{b}{a}\right)\implies \theta=tan^{-1}\left(\cfrac{b}{a}\right)\)