Download Trigonometric Form of a Complex Number

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PRECALCULUS NOTES & EXAMPLES
Section 6.5 Trigonometric Form of Complex Numbers
Objectives: Plot complex numbers in the complex plane
Write the trigonometric forms of complex numbers
The complex number z = a + b i can be represented by a point on the complex plane.
The absolute value of the complex number z = a + b i is the distance between the origin and
point (a, b).
Ex. Plot the complex number z = 3 – 4 i and find its absolute value.
The trigonometric form of a complex number is also called the polar form.
Because there are infinitely many choices for θ, the trigonometric form of a complex number is not
unique. Normally, θ is restricted to the interval 0 < θ < 2 π.
Represent the complex number graphically, and find the trigonometric form of the number.
1.
z = 1 − 3i
2.
z = −3 +
⎛
2π
Find the standard form of the complex number. z = 8 ⎜ cos
3
⎝
3 i
+ i sin
2π ⎞
3 ⎟⎠
Multiplication and Division of Complex Numbers
Suppose you are given two complex numbers
z1 = r1 (cosθ1 + i sin θ1 )
and
z2 = r2 (cosθ 2 + i sin θ 2 )
Find the product z1 z2 and simplify using sum and difference formulas for sine and cosine.
z1 z2 =
Product and Quotient of Two Complex Numbers
Let z1 = r1 (cosθ1 + i sin θ1 )
z1 z2 =
z1
=
z2
and
z2 = r2 (cosθ 2 + i sin θ 2 ) be complex numbers.
Ex. Find the product of the two complex numbers.
z1 = 2 (cos
2π
2π
+ i sin )
3
3
z2 = 8 (cos
11π
11π
+ i sin
)
6
6
Ex. Find the quotient of the two complex numbers.
z1 = 24 (cos 300 0 + i sin 300 0 )
z1 = 8 (cos 75 0 + i sin 75 0 )
Use the multiplication rule to find the following:
z = r (cosθ + i sin θ )
z2 =
z3 =
z4 =
.
.
.
zn =
DeMoivre’s Theorem
If z = r (cos θ + i sin θ ) is a complex number and n is a positive integer, then
zn =
(
Ex. Use DeMoivre’s to find −1 +
3i
)
12
.