Polar Form of Complex Number/Examples/4 cis 315

Example of Polar Form of Complex Number

The complex number $\polar {4, 315 \degrees}$ can be expressed in Cartesian form as:

$4 \cis 315 \degrees = 2 \sqrt 2 - 2 \sqrt 2 i$

and depicted in the complex plane as:

Proof

\(\ds 4 \cis 315 \degrees\)

\(=\)

\(\ds 4 \paren {\cos 315 \degrees + i \sin 315 \degrees}\)

\(\ds \)

\(=\)

\(\ds 4 \times \paren {\dfrac {\sqrt 2} 2 + \dfrac {-\sqrt 2} 2 i}\)

Cosine of $315 \degrees$ and Sine of $315 \degrees$

\(\ds \)

\(=\)

\(\ds 2 \sqrt 2 - 2 \sqrt 2 i\)

$\blacksquare$

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: $84 \ \text {(c)}$