Trigonometry
Ch6Sec6Day3 1 1/26/2019
Section 6.6 – Complex Numbers in Polar Form – Day 3
Objective:
3) Given complex numbers in polar form, find roots.
Complex Roots
Let w be a given complex number, and let
2n
denote a positive integer. Any complex number z that satisfies the
equation
wzn
is called a complex nth root of w. So, solutions of
wz2
are called the complex square roots of w
and solutions of
wz3
are called the complex cube roots of w.
Theorem – Finding Complex Roots
Let
)sincos(rw i
be a complex number. If
,0w
there are n distinct complex nth roots of w,
given by the formula
,
n
k360
n
sin
n
k360
n
cosrz n
k
i
where
.1n,,2,1,0k
Example 9: Find the three complex cube roots of
.31 i
:31 i
rectangular coordinates
)3,1()y,x(
Quadrant IV
22 yxr
22 )3()1(
31
4
2
Find the reference angle
:
ref
x
y
tan 1
ref
1
3
tan 1
3tan 1
3tan 1
60
is in Quadrant IV, thus,
ref
360
60360
300
Thus, the polar form of
i31
is
)sincos(r31 ii
)300sin300cos(2 i
