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Yordan · Answer 1 · 2023-03-08T08:07:09+0000

A complex number z is given in the form:

z = (x.y) = (realpart) x + imaginary part (iy)

In this case:

z1 = -7i

z2 = 3+3i

To write in trigonometric form:

$\begin{gathered} z\text{ = r\lparen cos}\theta\text{ + isin}\theta) \\ For\text{ z1} \\ r\text{ = }√(0^2+7^2) \\ \text{ = 7} \\ \theta\text{ =}\tan^(-1)((7)/(0) \\ Since\text{ t}he\text{ }argument\text{ }is\text{ }undefined\text{ }and\text{ y is negative,} \\ \theta=\text{ }(3\pi)/(2) \\ In\text{ trig form:} \\ z1\text{ = 7\lparen cos}(3\pi)/(2);sin(3\pi)/(2)) \\ For\text{ z2} \\ r\text{ = }\sqrt{3^2\text{ +3}^2} \\ \text{ =3}√(2) \\ \theta\text{ = }\tan^(-1)(3)/(3) \\ =\text{ }(\pi)/(4) \\ In\text{ trig form:} \\ z2\text{ = 3}√(2)(cos(\pi)/(4);sin(\pi)/(4)) \end{gathered}$

Multiplication in trigonometric form:

$z1*z2\text{ = \lparen21}√(2)\text{ \rparen \lparen cos}(7\pi)/(4);\text{ sin}(7\pi)/(4))$

Multiplication in standard form:

$\begin{gathered} (-7i)(3\text{ + 3i\rparen} \\ =-21i\text{ - 21i}^2 \\ i^2\text{ = -1} \\ =\text{ -21i + 21} \\ r\text{ = }√(21^2+21^2) \\ =21√(2) \end{gathered}$

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