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Question 1404c
Complex Numbers
Polar Form Arithmetic
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The complex number
z
{z}
z
is given by
z
=
γ
α
5
β
∗
,
z=\frac{\gamma}{\alpha^5 \beta^*},
z
=
α
5
β
∗
γ
,
where
α
=
2
(
cos
2
3
π
+
i
sin
2
3
π
)
,
β
=
3
(
cos
1
3
π
+
i
sin
1
3
π
)
,
γ
=
2
(
cos
(
−
2
3
π
)
+
i
sin
(
−
2
3
π
)
)
.
\begin{align*} \alpha &= { \textstyle 2 \left( \cos \frac{2}{3} \pi + \mathrm{i} \sin \frac{2}{3} \pi \right)}, \\ \beta &= { \textstyle 3 \left( \cos \frac{1}{3} \pi + \mathrm{i} \sin \frac{1}{3} \pi \right)}, \\ \gamma &= { \textstyle 2 \left( \cos \left( - \frac{2}{3} \pi \right) + \mathrm{i} \sin \left( - \frac{2}{3} \pi \right) \right)}. \end{align*}
α
β
γ
=
2
(
cos
3
2
π
+
i
sin
3
2
π
)
,
=
3
(
cos
3
1
π
+
i
sin
3
1
π
)
,
=
2
(
cos
(
−
3
2
π
)
+
i
sin
(
−
3
2
π
)
)
.
Find
∣
z
∣
{|z|}
∣
z
∣
and
k
,
{k,}
k
,
where
arg
(
z
)
=
k
π
{\arg(z)=k\pi}
ar
g
(
z
)
=
kπ
and
−
π
<
arg
(
z
)
<
π
.
{-\pi < \arg(z) < \pi.}
−
π
<
ar
g
(
z
)
<
π
.
Attempt
∣
z
∣
=
{|z| = }
∣
z
∣
=
k
=
{k = }
k
=
Answer
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