Polar Form of Complex Number/Examples/2 + 2 root 3 i
Jump to navigation
Jump to search
Example of Polar Form of Complex Number
The complex number $2 + 2 \sqrt 3 i$ can be expressed as a complex number in polar form as $\polar {4, \dfrac \pi 3}$.
Proof
| \(\ds \cmod {2 + 2 \sqrt 3 i}\) | \(=\) | \(\ds \sqrt {2^2 + \paren {2 \sqrt 3}^2}\) | Definition of Complex Modulus | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {4 + 4 \times 3}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {16}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 4\) |
Then:
| \(\ds \map \cos {\map \arg {2 + 2 \sqrt 3 i} }\) | \(=\) | \(\ds \dfrac 2 4\) | Definition of Argument of Complex Number | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \arg {2 + 2 \sqrt 3 i}\) | \(=\) | \(\ds \pm \dfrac \pi 3\) | Cosine of $60 \degrees$, Cosine of $300 \degrees$ |
| \(\ds \map \sin {\map \arg {2 + 2 \sqrt 3 i} }\) | \(=\) | \(\ds \dfrac {2 \sqrt 3} 4\) | Definition of Argument of Complex Number | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\sqrt 3} 2\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \arg {2 + 2 \sqrt 3 i}\) | \(=\) | \(\ds \dfrac \pi 3 \text { or } \dfrac {2 \pi} 3\) | Sine of $60 \degrees$, Sine of $120 \degrees$ |
Hence:
- $\map \arg {2 + 2 \sqrt 3 i} = \dfrac \pi 3$
and hence the result.
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Polar Form of Complex Numbers: $16 \ \text {(a)}$