Polar Form of Complex Number/Examples/5 cis 7 pi 6^-1

Example of Polar Form of Complex Number

The complex number $\polar {5, \dfrac {7 \pi} 6}$ can be expressed in Cartesian form as:

$5 \cis \dfrac {7 \pi} 6 = -5 \dfrac {\sqrt 3} 2 - \dfrac 5 2 i$

and depicted in the complex plane as:

Proof

\(\ds 5 \cis \dfrac {7 \pi} 6\)

\(=\)

\(\ds 5 \paren {\cos \dfrac {7 \pi} 6 + i \sin \dfrac {7 \pi} 6}\)

\(\ds \)

\(=\)

\(\ds 5 \times \paren {-\dfrac {\sqrt 3} 2 + -\dfrac 1 2 i}\)

Cosine of $210 \degrees$ and Sine of $210 \degrees$

\(\ds \)

\(=\)

\(\ds -5 \dfrac {\sqrt 3} 2 - \dfrac 5 2 i\)

$\blacksquare$

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