Polar Form of Complex Number/Examples/-5 + 5i

Example of Polar Form of Complex Number

The complex number $-5 + 5 i$ can be expressed as a complex number in polar form as $\polar {5 \sqrt 2, \dfrac {3 \pi} 4}$.

Proof

\(\ds \cmod {-5 + 5 i}\)

\(=\)

\(\ds \sqrt {\paren {-5}^2 + 5^2}\)

Definition of Complex Modulus

\(\ds \)

\(=\)

\(\ds \sqrt {2 \times 25}\)

\(\ds \)

\(=\)

\(\ds 5 \sqrt 2\)

Then:

\(\ds \map \cos {\map \arg {-5 + 5 i} }\)

\(=\)

\(\ds \dfrac {-5} {5 \sqrt 2}\)

Definition of Argument of Complex Number

\(\ds \)

\(=\)

\(\ds -\frac {\sqrt 2} 2\)

\(\ds \leadsto \ \ \)

\(\ds \map \arg {-5 + 5 i}\)

\(=\)

\(\ds \pm \dfrac {3 \pi} 4\)

Cosine of $135 \degrees$, Cosine of $225 \degrees$

\(\ds \map \sin {\map \arg {-5 + 5 i} }\)

\(=\)

\(\ds \dfrac 5 {5 \sqrt 2}\)

Definition of Argument of Complex Number

\(\ds \)

\(=\)

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

\(\ds \leadsto \ \ \)

\(\ds \map \arg {-5 + 5 i}\)

\(=\)

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

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

Hence:

$\map \arg {-5 + 5 i} = \dfrac {3 \pi} 4$

and hence the result.

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Polar Form of Complex Numbers: $16 \ \text {(b)}$