Polar Form of Complex Number/Examples/2 root 2 + 2 root 2 i

Example of Polar Form of Complex Number

The complex number $2 \sqrt 2 + 2 \sqrt 2 i$ can be expressed as a complex number in polar form as $\polar {4, \dfrac \pi 4}$.

Proof

\(\ds \cmod {2 \sqrt 2 + 2 \sqrt 2 i}\)

\(=\)

\(\ds \sqrt {\paren {2 \sqrt 2}^2 + \paren { 2 \sqrt 2}^2}\)

Definition of Complex Modulus

\(\ds \)

\(=\)

\(\ds \sqrt {8 + 8}\)

\(\ds \)

\(=\)

\(\ds \sqrt {16}\)

\(\ds \)

\(=\)

\(\ds 4\)

Then:

\(\ds \map \cos {\map \arg {2 \sqrt 2 + 2 \sqrt 2 i} }\)

\(=\)

\(\ds \dfrac {2 \sqrt 2} 4\)

Definition of Argument of Complex Number

\(\ds \)

\(=\)

\(\ds \dfrac {\sqrt 2} 2\)

\(\ds \leadsto \ \ \)

\(\ds \map \arg {2 \sqrt 2 + 2 \sqrt 2 i}\)

\(=\)

\(\ds \dfrac \pi 4 \text { or } \dfrac {7 \pi} 4\)

Cosine of $45 \degrees$, Cosine of $315 \degrees$

\(\ds \map \sin {\map \arg {2 \sqrt 2 + 2 \sqrt 2 i} }\)

\(=\)

\(\ds \dfrac {2 \sqrt 2} 4\)

Definition of Argument of Complex Number

\(\ds \)

\(=\)

\(\ds \dfrac {\sqrt 2} 2\)

\(\ds \leadsto \ \ \)

\(\ds \map \arg {2 \sqrt 2 + 2 \sqrt 2 i}\)

\(=\)

\(\ds \dfrac \pi 4 \text { or } \dfrac {3 \pi} 4\)

Sine of $45 \degrees$, Sine of $135 \degrees$

Hence:

$\map \arg {2 \sqrt 2 + 2 \sqrt 2 i} = \dfrac \pi 4$

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: $81 \ \text {(c)}$