Polar Form of Complex Number/Examples/-3 - 4i

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Example of Polar Form of Complex Number

The complex number $-3 - 4 i$ can be expressed as a complex number in polar form as $\polar {5, \pi + \arctan {\dfrac 4 3} }$.


Proof

\(\ds \cmod {-3 - 4 i}\) \(=\) \(\ds \sqrt {\paren {-3}^2 + \paren {-4}^2}\) Definition of Complex Modulus
\(\ds \) \(=\) \(\ds \sqrt {9 + 16}\)
\(\ds \) \(=\) \(\ds \sqrt {25}\)
\(\ds \) \(=\) \(\ds 5\)


Then:

\(\ds \map \cos {\map \arg {-3 - 4 i} }\) \(=\) \(\ds \dfrac {-3} 5\) Definition of Argument of Complex Number
\(\ds \leadsto \ \ \) \(\ds \map \arg {-3 - 4 i}\) \(=\) \(\ds \map \arccos {-\dfrac 3 5}\)


\(\ds \map \sin {\map \arg {-3 - 4 i} }\) \(=\) \(\ds \dfrac {-4} 5\) Definition of Argument of Complex Number
\(\ds \leadsto \ \ \) \(\ds \map \arg {-3 - 4 i}\) \(=\) \(\ds \map \arccos {-\dfrac 4 5}\)


\(\ds \map \tan {\map \arg {-3 - 4 i} }\) \(=\) \(\ds \frac {\map \sin {\map \arg {-3 - 4 i} } } {\map \cos {\map \arg {-3 - 4 i} } }\) Definition of Tangent Function
\(\ds \) \(=\) \(\ds \dfrac {-4 / 5} {- 3 / 5}\)
\(\ds \) \(=\) \(\ds \dfrac 4 3\)

Hence:

$\map \arg {-3 - 4 i} = \pi + \arctan {\dfrac 4 3}$

as $-3 - 4 i$ is in Quadrant III.

$\blacksquare$


Sources