Download Algebra Expressions and Real Numbers

Section 7.5
Complex Numbers in Polar Form:
DeMoive’s Theorem
The Complex Plane
Example
Determine the absolute value of each of the following:
a. z=2+4i
b. z=1-6i
c. z=7i
Polar Form of a
Complex Number
Example
Plot z=3-2i in the complex plane. Then write z
Imaginary
in polar form.
Axis
Real
Axis
Example
Write z = 2 (cos 3000 + i sin3000 ) in rectangular form.
Imaginary
Axis
Real
Axis
Products and Quotients in Polar
Form
Example
Find the product z1z 2 if z1  3 (cos 450  i sin 450 ),
and z 2  2 (cos1350  i sin1350 ).
Example
z2
Find the quotient
if z1  3 (cos 450  i sin 450 ),
z1
and z 2  2 (cos1350  i sin1350 ).
Example
z2
Find the quotient
if z1  10 (cos 600  i sin  600 ),
z1
and z 2  5 (cos1500  i sin1500 ).
Powers of Complex Numbers in
Polar Form
Example
Find z if z  3(cos 45  i sin 45 ).
5
0
0
Example
Find z if z  2(cos 60  i sin 60 ).
3
0
0
Roots of Complex Numbers in
Polar Form
Example
Find all 5th roots of 1=1 (cos 00  i sin 00 )
y
x





Example
Find all 4th roots of -8+8i 3=16 (cos1200  i sin 1200 )
y
x





Write the complex number z=- 3  i in polar form.
0
0
z

2(cos
30
+i
sin30
)
(a)
(b) z  2(cos150 +i sin150 )
0
0
(c) z  2(cos 600 +i sin600 )
(d) z  2(cos 2100 +i sin2100 )
Write z=2(cos 3000  i sin 3000 ) in rectangular form.
(a) z  1  i 2
1
i 3
2
(c) z  1  i 3
(b) z 
(d) z  2  i 3