Polar Form of Complex Number/Examples/2 cis 5 pi 4^-1

Example of Polar Form of Complex Number

The complex number $\polar {2, \dfrac {5 \pi} 4}$ can be expressed in Cartesian form as:

$2 \cis \dfrac {5 \pi} 4 = -\sqrt 2 - \sqrt 2 i$

and depicted in the complex plane as:

Proof

\(\ds 2 \cis \dfrac {5 \pi} 4\)

\(=\)

\(\ds 2 \paren {\cos \dfrac {5 \pi} 4 + i \sin \dfrac {5 \pi} 4}\)

\(\ds \)

\(=\)

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

Cosine of $225 \degrees$ and Sine of $225 \degrees$

\(\ds \)

\(=\)

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

$\blacksquare$

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