Polar Form of Complex Number/Examples/-2 root 3 - 2 i
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Example of Polar Form of Complex Number
The complex number $-2 \sqrt 3 - 2 i$ can be expressed as a complex number in polar form as $\polar {4, \dfrac {7 \pi} 6}$.
Proof
\(\ds \cmod {-2 \sqrt 3 - 2 i}\) | \(=\) | \(\ds \sqrt {\paren {-2 \sqrt 3}^2 + \paren {-2}^2}\) | Definition of Complex Modulus | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {4 \times 3 + 4}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {16}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4\) |
Then:
\(\ds \map \cos {\map \arg {-2 \sqrt 3 - 2 i} }\) | \(=\) | \(\ds \dfrac {-2 \sqrt 3} 4\) | Definition of Argument of Complex Number | |||||||||||
\(\ds \) | \(=\) | \(\ds -\dfrac {\sqrt 3} 2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \arg {-2 \sqrt 3 - 2 i}\) | \(=\) | \(\ds \dfrac {5 \pi} 6 \text { or } \dfrac {7 \pi} 6\) | Cosine of $150 \degrees$, Cosine of $210 \degrees$ |
\(\ds \map \sin {\map \arg {-2 \sqrt 3 - 2 i} }\) | \(=\) | \(\ds \dfrac {-2} 4\) | Definition of Argument of Complex Number | |||||||||||
\(\ds \) | \(=\) | \(\ds -\dfrac 1 2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \arg {-2 \sqrt 3 - 2 i}\) | \(=\) | \(\ds \dfrac {7 \pi} 6 \text { or } \dfrac {11 \pi} 6\) | Sine of $210 \degrees$, Sine of $330 \degrees$ |
Hence:
- $\map \arg {-2 \sqrt 3 - 2 i} = \dfrac {7 \pi} 6$
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: $81 \ \text {(f)}$