Transcript Advanced Precalculus Notes 8.3 The Complex Plane: De

Advanced Precalculus Notes 8.3 The
Complex Plane: De Moivre’s Theorem
Absolute value of a number: distance from zero
(origin)
Complex number: z = a + bi
Conjugate: z = a - bi
Magnitude or modulus:
| z | x  y  z z
2
2
Argument:
b
  tan
a
1
Polar Form of a Complex number:
z  x  yi  (r cos )  (r sin  )i  r (cos  i sin  )  rcis
Plot the point corresponding to z  3  i
in the complex plane and write it in polar form:
Plot the point corresponding to z  2(cos30  i sin 30)
in the complex plane and write it in rectangular form:
Product of complex numbers:
z1 z2  r1r2 [cos(1  2 )  i sin(1  2 )]
Quotient of complex numbers:
z1 r1
 [cos(1   2 )  i sin(1   2 )]
z2 r2
Given: z  3(cos20  i sin 20) and w  5(cos100  i sin 100)
find: a) zw
b)
z
w
DeMoivre’s Theorem:
z n  r n [cos(n )  i sin(n )]
Write
[2(cos20  i sin 20)]3
in standard form a + bi
Write (1  i)
in standard form a + bi
5
Complex Roots:
  0 2k 
  0 2k 
zk  r cos 
  i sin 

n 
n 
n
 n
n
Find the complex cube roots of:
in polar form and standard form
 1 3i
Assignment:
page 606:
1 – 11, 16, 19, 23, 27, 33,
41, 43, 53, 57