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11
Complex #'s Fun
Complex #'s Fun
Source:
December 21, 2005
trigonometry
AMC
AIME
complex numbers
Cyclotomic Polynomials
Problem Statement
Let
P
P
P
be the product of the roots of
z
6
+
z
4
+
z
3
+
z
2
+
1
=
0
z^6+z^4+z^3+z^2+1=0
z
6
+
z
4
+
z
3
+
z
2
+
1
=
0
that have positive imaginary part, and suppose that
P
=
r
(
cos
θ
∘
+
i
sin
θ
∘
)
,
P=r(\cos \theta^\circ+i\sin \theta^\circ),
P
=
r
(
cos
θ
∘
+
i
sin
θ
∘
)
,
where
0
<
r
0<r
0
<
r
and
0
≤
θ
<
360.
0\le \theta <360.
0
≤
θ
<
360.
Find
θ
.
\theta.
θ
.
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