Polar Form of Complex Number/Examples/-1 + root 3 i

Example of Polar Form of Complex Number

The complex number $-1 + \sqrt 3 i$ can be expressed as a complex number in polar form as $\polar {2, \dfrac {2 \pi} 3}$.

$\ds \cmod {-1 + \sqrt 3 i}$

$=$

$\ds \sqrt {\paren {-1}^2 + \paren {\sqrt 3}^2}$

$\ds $

$=$

$\ds \sqrt {1 + 3}$

$\ds $

$=$

$\ds \sqrt 4$

$\ds $

$=$

$\ds 2$

Then:

$\ds \map \cos {\map \arg {-1 + \sqrt 3 i} }$

$=$

$\ds \dfrac {-1} 2$

$\ds \leadsto \ \ $

$\ds \map \arg {-1 + \sqrt 3 i}$

$=$

$\ds \dfrac {2 \pi} 3 \text { or } \dfrac {4 \pi} 3$

$\ds \map \sin {\map \arg {-1 + \sqrt 3 i} }$

$=$

$\ds \dfrac {\sqrt 3} 2$

$\ds \leadsto \ \ $

$\ds \map \arg {-1 + \sqrt 3 i}$

$=$

$\ds \dfrac \pi 3 \text { or } \dfrac {2 \pi} 3$

Hence:

$\map \arg {-1 + \sqrt 3 i} = \dfrac {2 \pi} 3$

and hence the result.

$\blacksquare$

1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Polar Form of Complex Numbers: $81 \ \text {(b)}$