Download Nov 14 Trigonometric Form of Complex Numbers, Quiz 5 Homework

Nov 14
Trigonometric Form of Complex Numbers, Quiz 5
Homework: 13.6 #5, 11, 23, 31, 37, 45, 49, 55, 69, 71
define modulus, argument and, trigonometric form of complex number. Complex
numbers may be located in the complex plane by defining their distance
from the origin and the angle (formed by a line segment from the origin to
the point) from the positive x-axis (real axis). The distance of a complex
number z = a + bi is z = a 2 + b 2 and is called the modulus of z. The
angle ! of z may be found by determining the arctangent of b a and is
called the argument of z. We can write any complex number in
trigonometric (or polar) form as z = z (cos! + isin ! ) , where ! is the
argument of z and z is the modulus of z. Write z = 3 + i in trig form
!
!%
"
and plot z in the complex plane. Plot z = 2 $ cos + i sin ' in the
#
2
2&
complex plane and convert z to rectangular form.
3 1
multiply complex numbers in trig form. Multiply z =
+ i by w = 1 + 3i
2 2
geometrically (in the complex plane).
! 3 1 $
3 1
+ i& =
+ i . Plot
1. First multiply z by the real part of w, 1#
2 2
" 2 2 %
3 1
+ i . Multiplying a complex number z by a real number w
2 2
stretches z by a factor of a . (If a < 0 , you also rotate by 180° .)
2. Now multiply z by the imaginary part of w ( 3i ),
! 3 1 $
3 3
3 3
3i #
+ i& = '
+ i . Plot !
+ i . The resulting product
2 2
2 2
" 2 2 %
is obtained from z by rotating 90° counterclockwise and then
stretching by a factor of b .
3. Now add the complex numbers
3 1
3 3
+ i and !
+ i to get
2 2
2 2
0 + 2i = 2i . Plot 0 + 2i = 2i .
4. Looking at the trigonometric forms of z and w [ z = ( cos 30° + i sin 30° )
and w = 2 ( cos 60° + i sin 60° ) ], we can write the product wz as
wz = 0 + 2i = 2 ( cos 90° + i sin 90° ) . In general, the argument of a
product, wz, of two complex numbers is equal to the sum of the
arguments of w and z, and the length of the product wz is equal to the
product of the lengths of w and z. To find the product of two complex
numbers in trig form, we multiply their moduli and add their
arguments.
divide complex numbers in trig form. There is a similar rule for dividing
complex numbers in trigonometric (polar) form: Divide their moduli and
subtract their arguments. Let z = !
2
2
1
3
+
i and w = +
i . Find
2
2
2 2
z w.
state DeMoivre's theorem (finding powers of complex numbers in trig form). We
can extend our earlier result about multiplying complex numbers in trig
form to find powers of complex numbers. This result known as
n
DeMoivre's Theorem states that z n = z (cos n! + isin n! ) , i.e., the length
of z n is the length of z raised to the nth power and the argument of z n is n
"
!
!%
times the argument of z. Let z = 2$ cos + isin ' . Find z 3 .
#
3
3&
use DeMoivre's theorem to find nth roots of complex numbers. We can use
DeMoivre's Theorem to find nth roots of complex numbers, too. Find the
three cube roots of !8 .