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
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Polar Form of Complex Numbers: $83 \ \text {(a)}$