Product of Complex Numbers in Polar Form/Examples/5 cis 20 x 3 cis 40

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Example of Use of Product of Complex Numbers in Polar Form

$5 \cis 20 \degrees \times 3 \cis 40 \degrees = \dfrac {15} 2 + \dfrac {15 \sqrt 3} 2 i$


Proof

\(\ds 5 \cis 20 \degrees \times 3 \cis 40 \degrees\) \(=\) \(\ds \paren {5 \times 3} \, \map \cis {20 \degrees + 40 \degrees}\) Product of Complex Numbers in Polar Form
\(\ds \) \(=\) \(\ds 15 \cis 60 \degrees\)
\(\ds \) \(=\) \(\ds 15 \paren {\cos 60 \degrees + i \sin 60 \degrees}\)
\(\ds \) \(=\) \(\ds 15 \times \paren {\dfrac 1 2} + 15 i \paren {\dfrac {\sqrt 3} 2}\) Cosine of $60 \degrees$, Sine of $60 \degrees$
\(\ds \) \(=\) \(\ds \dfrac {15} 2 + \dfrac {15 \sqrt 3} 2 i\) tidying

$\blacksquare$


Sources