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Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Objective:β’ Plot complex number in the complex plane.
β’ Find the absolute value of a complex number.
β’ Write complex numbers in polar form.
β’ Convert a complex number from polar to rectangular form.
β’ Find products of complex numbers in polar form.
β’ Find powers of complex numbers in polar form.
β’ Find roots of complex numbers in polar form.
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
A complex number is represented as a point in a coordinate plane. The horizontal axis of the coordinate plane is called the real axis.
The vertical axis is called the imaginary axis.
The coordinate system is called the complex plane.
When we represent a complexnumber as a point in the complexplane, we say that we areplotting the complex number.
The Complex Plane
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Plot the complex number in the complex planea)
z a bi 2 3z i 2, 3a b
We plot the point (a, b) = (2, 3).
2 3z i
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Plot the complex number in the complex planeb)
z a bi 3 5z i 3, 5a b
We plot the point (a, b) = (β3, β5).
3 5z i
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Plot the complex number in the complex planea)
z a bi 4 0z i 4, 0a b
We plot the point (a, b) = (β4, 0).
4 0z i
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Plot the complex number in the complex planea)
z a bi 0z i 0, 1a b
We plot the point (a, b) = (0, β1).
0z i
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
The Absolute Value of a Complex NumberThe absolute value of the complex number is the distance from the origin to the point in the complex plane.The absolute value of the complex number is
|π|=|π+ππ|=βππ+ππ
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Determine the absolute value of the following complex number: a)
2 2 .z a bi a b
2 25 12 5 12z i
25 144
16913
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Determine the absolute value of the following complex number: b)
2 2 .z a bi a b 2 22 3 2 ( 3)z i
4 9
13
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Polar Form of a Complex NumberA complex number in the form is said to be in rectangular form.
The expression is called the polar form of a complex number.
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Plot the complex number in the complex plane, then write the number in polar form:
z a bi
1 3z i
1, 3a b
1 3z i
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Plot the complex number in the complex plane, then write the number in polar form:
1 3z i
2 2r a b
22( 1) 3
1 3 4 2
tanba
3
31
43
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Plot the complex number in the complex plane, then write the number in polar form:
1 3z i
The polar form of is
42,
3r
(cos sin )z r i
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Write in rectangular form.
The complex number is in polar form, with and . We use exact values for cos 60Β° and sin 60Β° to write the numberin rectangular form.
2 (cosπ+π sin π )ΒΏ2( 12 +π β32 )
ΒΏπ+πβπThe rectangular form of is
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Products and Quotients in Polar Form
We can multiply and divide complex numbers fairly quickly if the numbers are expressed in polar form.
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Find the product of the complex numbers. Leave the answer in polar form.
π§1=6 (cos 40 Β°+π sin 40 Β° ) π§ 2=5 (cos 20 Β°+ π sin 20 Β° )
ππ ππ=π πππ [πππ (π½π+π½π )+π πππ (π½π+π½π ) ]ππ ππ=π βπ [πππ (ππΒ°+ππΒ° )+ππππ (ππΒ°+ππΒ° ) ]
ΒΏππ (ππ¨π¬ππΒ°+ππ¬π’π§ππΒ° )
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Find quotients of complex numbers in polar form.
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Find the quotient of the following complex numbers. Leave the answer in polar form.
π§1=50(cos 4π3 +π sin 4π3 ) π§ 2=5(cos π3 +π sin π
3 )1 1
1 2 1 22 2
[(cos( ) sin( )]z r
iz r
1
2
50 4 4cos sin
5 3 3 3 3z
iz
10(cos sin )i
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Powers of Complex Numbers in Polar Form
The formula for the nth power of a complex number is known as DeMoivreβs Theorem in honor of the French mathematician Abraham DeMoivre (1667β1754).
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Find . Write the answer in rectangular form, a + bi .
ππ=ππ (ππ¨π¬ππ½+ππ¬π’π§ππ½ )
[2 (cos30 Β°+ πsin 30 Β° ) ]5ΒΏ25 [cos (5 β30 Β° )+π sin (5 β30 Β° ) ]ΒΏ32 [cos (150 Β° )+π sin (150 Β° ) ]
ΒΏ32 (β β32
+ 12π)
ΒΏβππ βπ+πππ
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Find , a + bi .
Solution DeMoivreβs Theorem applies to complex numbers in polar form.Thus, we must first write in form. Then we can useDeMoivreβs Theorem. The complex number is plotted in Figure 6.44. Fromthe figure, we obtain values for and .π=βπ2+π2ΒΏβ12+12ΒΏβ2
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Find , a + bi .
v
v
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Roots of Complex Numbers in Polar Form
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Roots of Complex Numbers in Polar Form
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
There are exactly four fourth roots of the given complex number. The four fourth roots are found by substituting 0, 1, 2, and 3 for k in the expression
ππ=πβπ [πππ( π½+πππΒ°π
π )+π πππ(π½+πππΒ°ππ )]
π§π=4β16 [πππ ( 120Β°+360 Β° βπ4 )+ππ ππ( 120 Β°+360 Β° βπ4 )]
Find all the complex fourth roots of . Write roots in polar form, with in degrees.
ΒΏ2(cos 120 Β°4 + πsin 120 Β°4 )
ΒΏπ (πππππΒ°+π πππππΒ° )
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
π§π=4β16[πππ (120 Β°+360 Β° βπ4 )+ ππ ππ(120 Β°+360 Β° βπ4 )]
ΒΏ2(cos 480 Β°4 +π sin 480 Β°4 )
ΒΏπ (ππππππΒ°+πππππππΒ° )
Find all the complex fourth roots of . Write roots in polar form, with in degrees.
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
π§π=4β16[πππ (120 Β°+360 Β° βπ4 )+ ππ ππ(120 Β°+360 Β° βπ4 )]
ΒΏ2(cos 840 Β°4 +π sin 840Β°4 )
ΒΏπ (ππππππΒ°+πππππππΒ° )
Find all the complex fourth roots of . Write roots in polar form, with in degrees.
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
π§π=4β16[πππ (120 Β°+360 Β° βπ4 )+ ππ ππ(120 Β°+360 Β° βπ4 )]
ΒΏ2(cos 1200 Β°4 + πsin 1200 Β°4 )
ΒΏπ (ππππππΒ°+πππππππΒ° )
Find all the complex fourth roots of . Write roots in polar form, with in degrees.
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Find all the complex fourth roots of . Write roots in polar form, with in degrees.
The four complex fourth roots are:
ππ=π (πππππΒ°+π πππππΒ° )ππ=π (ππππππΒ°+π ππππππΒ° )ππ=π (ππππππ Β°+π ππππππ Β° )ππ=π (ππππππ Β°+π ππππππ Β° )
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Find all the cube roots of 8. Write roots in rectangular form.
Solution DeMoivreβs Theorem for roots applies to complex numbers in polar form. Thus, we will first write 8, or in polar form. We express in radians, although degrees could also be used.
8=π (cosπ+ π sinπ )ΒΏ8 (cos0+ π sin 0 )
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Find all the cube roots of 8. Write roots in rectangular form.
The three cube roots of 8 are found by substituting 0, 1, and 2 for k in the expression for above the voice balloon. Thus, the three cube roots of 8 are
ππ=πβπ [πππ( π½+ππ Β°π
π )+π πππ(π½+ππ Β°ππ )]
π§π=3β8 [πππ ( 0 Β°+2π Β° βπ4 )+π π ππ( 0 Β°+2π Β° βπ4 ) ]
ΒΏ2 (cos 0 Β°+π sin 0 Β° )
ΒΏπΒΏ2 (1+π β0 )
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Find all the cube roots of 8. Write roots in rectangular form.
π§π=3β8 [πππ ( 0 Β°+2π Β° βπ4 )+ππ ππ( 0 Β°+2π Β° βπ4 )]
ΒΏ2(cos 2π3 +π sin 2π3 )
ΒΏβπ+πβπΒΏ2(β 12+ πβ β32 )
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Find all the cube roots of 8. Write roots in rectangular form.
π§π=3β8 [πππ ( 0 Β°+2π Β° βπ4 )+ππ ππ( 0 Β°+2π Β° βπ4 )]
ΒΏ2(cos 4 π3 +π sin 4π3 )
ΒΏβπβ πβπ
ΒΏ2(β 12+ πβ(β β32 ))
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Find all the cube roots of 8. Write roots in rectangular form.
Mrs. RivasHomework
Pg. 696-697 # 12-26 Even 30, 32, 42, 44, 46, 48, 52, 54, 58, 62, 66, 70, 72, 76