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Introduction to Complex
Numbers in Trigonometric Form
Accelerated Precalculus
•The unit imaginary number is i = √-1 .
•Remember that i 2 = -1 .
•A complex number is the sum of a real number
and an imaginary number.
•The general (standard) form of a complex
number is z = a + bi , where the real number a is
called the real part and the real number b is called
the imaginary part of z.
•The complex conjugate of a + bi is a – bi.
Complex Number Operations:
Let a + bi and c + di be any two complex numbers.
Addition: (a + bi ) + (c + di) = ( a + c ) + (b + d)i
(Add the real parts & add the imaginary)
Subtraction: (a + bi ) - (c + di) = ( a - c ) + (b - d)i
(Subtract the real parts & subtract the imaginary)
Complex Number Operations:
Multiplication: (a + bi )(c + di)
(a + bi )(c + di) = ac + adi +bci + bdi2
= (ac - bd) + (ad +bc)i
(Multiply like you would any binomial - FOIL )
Division: (a + bi ) / (c + di)
(a + bi) (c – di) = (ac + bd) + (-ad + bc)i
(c + di) (c – di)
c2 + d2
(Multiply numerator & denominator by the complex
conjugate of the denominator )

Geometric Representation of a complex number is
graphed in the complex plane as a point (a, b). The
x-axis is called the real axis and the y-axis is the
Imaginary axis
imaginary axis.
( -1, 2)
-1 + 2i
Distance from origin is
called the modulus.
z  = a + bi = √(a2+b2)
•
(1, √3 )
1 + √3 i
 z = 2
60°
1
Real axis
( 0, -2) •
0 - 2i
Trigonometric form of a complex number (z)
includes the modulus and angle formed with the
positive x-axis.
Convert complex numbers to their trig form
The right triangle determined by z = a + bi has
side lengths of a, b, and r (radius or modulus) :
r = √(a2+b2) cos θ = a/r
sin θ = b/r
a = r cos θ
b = r sin θ
Therefore a + bi = (r cos θ) + (r sin θ)i or
Imaginary
z = r (cos θ + i sin θ)
(a, b )
a+bi
r
b
θ
Real
a
θ is called the argument
θ = tan-1(b / a)
(usually in radians)
Product and Quotient of Trigonometric Form
Let z1= r1 (cos θ1 + i sin θ1 ) (abbreviated z1= r1cis θ1)
and z2= r2 (cos θ2+ i sin θ2 )
or z2= r2 cis θ2
z1 z2 = r1 r2 [cos (θ1 + θ2 )+ i sin (θ1 + θ2)]
(Multiply the moduli; add the arguments)
z1 = r1 [cos (θ1 - θ2 )+ i sin (θ1 - θ2)]
z2 r2
r2 ≠ 0
(Divide the moduli; subtract the arguments)
Assignment
Page 450-451: 1-39 odd Please remember
to write down each problem, show sketches
where necessary or helpful, and then show
each step of the solutions. Answers only will
not be accepted!