Product of Complex Numbers in Polar Form/Examples/3 cis 40 x 4 cis 80

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Example of Use of Product of Complex Numbers in Polar Form

$3 \cis 40 \degrees \times 4 \cis 80 \degrees = -6 + 6 \sqrt 3 i$


Proof

\(\ds 3 \cis 40 \degrees \times 4 \cis 80 \degrees\) \(=\) \(\ds \paren {3 \times 4} \map \cis {40 \degrees + 80 \degrees}\) Product of Complex Numbers in Polar Form
\(\ds \) \(=\) \(\ds 12 \cis 120 \degrees\)
\(\ds \) \(=\) \(\ds 12 \paren {\cos 120 \degrees + i \sin 120 \degrees}\)
\(\ds \) \(=\) \(\ds 12 \times \paren {-\dfrac 1 2} + 12 i \paren {\dfrac {\sqrt 3} 2}\) Cosine of $120 \degrees$, Sine of $120 \degrees$
\(\ds \) \(=\) \(\ds -6 + 6 \sqrt 3 i\) simplifying

$\blacksquare$


Sources