Polar Form of Complex Number/Examples/1 - 2i
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Example of Polar Form of Complex Number
The complex number $1 - 2 i$ can be expressed as a complex number in polar form as $\polar {\sqrt 5, -\arctan 2}$.
Proof
| \(\ds \cmod {1 - 2 i}\) | \(=\) | \(\ds \sqrt {1^2 + \paren {-2}^2}\) | Definition of Complex Modulus | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {1 + 4}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt 5\) |
Then:
| \(\ds \map \cos {\map \arg {1 - 2 i} }\) | \(=\) | \(\ds \dfrac 1 {\sqrt 5}\) | Definition of Argument of Complex Number | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \arg {1 - 2 i}\) | \(=\) | \(\ds \map \arccos {\dfrac 1 {\sqrt 5} }\) |
| \(\ds \map \sin {\map \arg {1 - 2 i} }\) | \(=\) | \(\ds \dfrac {-2} {\sqrt 5}\) | Definition of Argument of Complex Number | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \arg {1 - 2 i}\) | \(=\) | \(\ds \map \arccos {-\dfrac 2 {\sqrt 5} }\) |
| \(\ds \map \tan {\map \arg {1 - 2 i} }\) | \(=\) | \(\ds \frac {\map \sin {\map \arg {1 - 2 i} } } {\map \cos {\map \arg {1 - 2 i} } }\) | Definition of Tangent Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {-2 / {\sqrt 5} } {1 / {\sqrt 5} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -2\) |
Hence:
- $\map \arg {1 - 2 i} = \map \arctan {-2} = -\arctan 2$
as $1 - 2 i$ is in Quadrant II.
$\blacksquare$
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: $83 \ \text {(b)}$