Polar Form of Complex Number/Examples/6 cis 135

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Example of Polar Form of Complex Number

The complex number $\polar {6, 135 \degrees}$ can be expressed in Cartesian form as:

$6 \cis 135 \degrees = -3 \sqrt 2 + 3 \sqrt 2 i$

and depicted in the complex plane as:


6 cis 135.png


Proof

\(\ds 6 \cis 135 \degrees\) \(=\) \(\ds 6 \paren {\cos 135 \degrees + i \sin 135 \degrees}\)
\(\ds \) \(=\) \(\ds 6 \times \dfrac {-\sqrt 2} 2 + 6 i \times \dfrac {\sqrt 2} 2\) Cosine of $135 \degrees$ and Sine of $135 \degrees$
\(\ds \) \(=\) \(\ds -3 \sqrt 2 + 3 \sqrt 2 i\)

$\blacksquare$


Sources