Polar Form of Complex Number/Examples/-3i

Example of Polar Form of Complex Number

The complex number $-3 i$ can be expressed as a complex number in polar form as $\polar {3, \dfrac {3 \pi} 2}$.

Proof

\(\ds \cmod {-3 i}\)

\(=\)

\(\ds \sqrt {\paren {-3}^2 + 0^2}\)

Definition of Complex Modulus

\(\ds \)

\(=\)

\(\ds 3\)

Then:

\(\ds \map \cos {\map \arg {-3 i} }\)

\(=\)

\(\ds \dfrac 0 3\)

Definition of Argument of Complex Number

\(\ds \)

\(=\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds \map \arg {-3 i}\)

\(=\)

\(\ds \dfrac {\pi} 2 \text { or } \dfrac {3 \pi} 2\)

Cosine of Half-Integer Multiple of Pi

\(\ds \map \sin {\map \arg {-3 i} }\)

\(=\)

\(\ds \dfrac {-3} 3\)

Definition of Argument of Complex Number

\(\ds \)

\(=\)

\(\ds -1\)

\(\ds \leadsto \ \ \)

\(\ds \map \arg {-3 i}\)

\(=\)

\(\ds \dfrac {3 \pi} 2\)

Sine of Half-Integer Multiple of Pi

Hence:

$\map \arg {-3 i} = \dfrac {3 \pi} 2$

and hence the result.

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Polar Form of Complex Numbers: $16 \ \text {(d)}$