Polar Form of Complex Number/Examples/2 + i
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Example of Polar Form of Complex Number
The complex number $2 + i$ can be expressed as a complex number in polar form as $\polar {\sqrt 5, \arctan {\dfrac 1 2} }$.
Proof
| \(\ds \cmod {2 + i}\) | \(=\) | \(\ds \sqrt {2^2 + 1^2}\) | Definition of Complex Modulus | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {4 + 1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt 5\) |
Then:
| \(\ds \map \cos {\map \arg {2 + i} }\) | \(=\) | \(\ds \dfrac 2 {\sqrt 5}\) | Definition of Argument of Complex Number | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \arg {2 + i}\) | \(=\) | \(\ds \arccos \dfrac 2 {\sqrt 5}\) |
| \(\ds \map \sin {\map \arg {2 + i} }\) | \(=\) | \(\ds \dfrac 1 {\sqrt 5}\) | Definition of Argument of Complex Number | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \arg {2 + i}\) | \(=\) | \(\ds \arcsin \dfrac 1 {\sqrt 5}\) |
| \(\ds \map \tan {\map \arg {2 + i} }\) | \(=\) | \(\ds \frac {\map \sin {\map \arg {2 + i} } } {\map \cos {\map \arg {2 + i} } }\) | Definition of Tangent Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {1 / \sqrt 5 } {2 / \sqrt 5}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 2\) |
Hence:
- $\map \arg {2 + i} = \arctan {\dfrac 1 2}$
and hence the result.
$\blacksquare$
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: $82$