Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Objective:• Plot complex number in the complex plane.
• Find the absolute value of a complex number.
• Write complex numbers in polar form.
• Convert a complex number from polar to rectangular form.
• Find products of complex numbers in polar form.
• Find powers of complex numbers in polar form.
• 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) = (–3, –5).
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) = (–4, 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, –1).
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
|𝒛|=|𝒂+𝒃𝒊|=√𝒂𝟐+𝒃𝟐
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 60° and sin 60° to write the numberin rectangular form.
2 (cos𝜃+𝑖 sin 𝜃 )¿2( 12 +𝑖 √32 )
¿𝟏+𝒊√𝟑The 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.
𝑧1=6 (cos 40 °+𝑖 sin 40 ° ) 𝑧 2=5 (cos 20 °+ 𝑖 sin 20 ° )
𝒛𝟏 𝒛𝟐=𝒓 𝟏𝒓𝟐 [𝒄𝒐𝒔 (𝜽𝟏+𝜽𝟐 )+𝒊 𝒔𝒊𝒏 (𝜽𝟏+𝜽𝟐 ) ]𝒛𝟏 𝒛𝟐=𝟔 ∙𝟓 [𝒄𝒐𝒔 (𝟒𝟎°+𝟐𝟎° )+𝒊𝒔𝒊𝒏 (𝟒𝟎°+𝟐𝟎° ) ]
¿𝟑𝟎 (𝐜𝐨𝐬𝟔𝟎°+𝒊𝐬𝐢𝐧𝟔𝟎° )
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.
𝑧1=50(cos 4𝜋3 +𝑖 sin 4𝜋3 ) 𝑧 2=5(cos 𝜋3 +𝑖 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 DeMoivre’s Theorem in honor of the French mathematician Abraham DeMoivre (1667–1754).
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 .
𝒛𝒏=𝒓𝒏 (𝐜𝐨𝐬𝒏𝜽+𝒊𝐬𝐢𝐧𝒏𝜽 )
[2 (cos30 °+ 𝑖sin 30 ° ) ]5¿25 [cos (5 ∙30 ° )+𝑖 sin (5 ∙30 ° ) ]¿32 [cos (150 ° )+𝑖 sin (150 ° ) ]
¿32 (− √32
+ 12𝑖)
¿−𝟏𝟔 √𝟑+𝟏𝟔𝒊
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
Find , a + bi .
Solution DeMoivre’s Theorem applies to complex numbers in polar form.Thus, we must first write in form. Then we can useDeMoivre’s Theorem. The complex number is plotted in Figure 6.44. Fromthe figure, we obtain values for and .𝑟=√𝑎2+𝑏2¿√12+12¿√2
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
𝒛𝒌=𝒏√𝒓 [𝒄𝒐𝒔( 𝜽+𝟑𝟔𝟎°𝒌
𝒏 )+𝒊 𝒔𝒊𝒏(𝜽+𝟑𝟔𝟎°𝒌𝒏 )]
𝑧𝟎=4√16 [𝑐𝑜𝑠( 120°+360 ° ∙𝟎4 )+𝑖𝑠𝑖𝑛( 120 °+360 ° ∙𝟎4 )]
Find all the complex fourth roots of . Write roots in polar form, with in degrees.
¿2(cos 120 °4 + 𝑖sin 120 °4 )
¿𝟐 (𝒄𝒐𝒔𝟑𝟎°+𝒊 𝒔𝒊𝒏𝟑𝟎° )
Mrs. Rivas
International Studies Charter
School.
Pre-Calculus
Section 6-5
COMPLEX NUMBERS IN POLAR FORM
𝑧𝟏=4√16[𝑐𝑜𝑠 (120 °+360 ° ∙𝟏4 )+ 𝑖𝑠𝑖𝑛(120 °+360 ° ∙𝟏4 )]
¿2(cos 480 °4 +𝑖 sin 480 °4 )
¿𝟐 (𝒄𝒐𝒔𝟏𝟐𝟎°+𝒊𝒔𝒊𝒏𝟏𝟐𝟎° )
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
𝑧𝟐=4√16[𝑐𝑜𝑠 (120 °+360 ° ∙𝟐4 )+ 𝑖𝑠𝑖𝑛(120 °+360 ° ∙𝟐4 )]
¿2(cos 840 °4 +𝑖 sin 840°4 )
¿𝟐 (𝒄𝒐𝒔𝟐𝟏𝟎°+𝒊𝒔𝒊𝒏𝟐𝟏𝟎° )
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
𝑧𝟑=4√16[𝑐𝑜𝑠 (120 °+360 ° ∙𝟑4 )+ 𝑖𝑠𝑖𝑛(120 °+360 ° ∙𝟑4 )]
¿2(cos 1200 °4 + 𝑖sin 1200 °4 )
¿𝟐 (𝒄𝒐𝒔𝟑𝟎𝟎°+𝒊𝒔𝒊𝒏𝟑𝟎𝟎° )
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:
𝒛𝟎=𝟐 (𝒄𝒐𝒔𝟑𝟎°+𝒊 𝒔𝒊𝒏𝟑𝟎° )𝒛𝟏=𝟐 (𝒄𝒐𝒔𝟏𝟐𝟎°+𝒊 𝒔𝒊𝒏𝟏𝟐𝟎° )𝒛𝟐=𝟐 (𝒄𝒐𝒔𝟐𝟏𝟎 °+𝒊 𝒔𝒊𝒏𝟐𝟏𝟎 ° )𝒛𝟑=𝟐 (𝒄𝒐𝒔𝟑𝟎𝟎 °+𝒊 𝒔𝒊𝒏𝟑𝟎𝟎 ° )
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 DeMoivre’s 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=𝑟 (cos𝜃+ 𝑖 sin𝜃 )¿8 (cos0+ 𝑖 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
𝒛𝒌=𝒏√𝒓 [𝒄𝒐𝒔( 𝜽+𝟐𝝅 °𝒌
𝒏 )+𝒊 𝒔𝒊𝒏(𝜽+𝟐𝝅 °𝒌𝒏 )]
𝑧𝟎=3√8 [𝑐𝑜𝑠( 0 °+2𝜋 ° ∙𝟎4 )+𝑖 𝑠𝑖𝑛( 0 °+2𝜋 ° ∙𝟎4 ) ]
¿2 (cos 0 °+𝑖 sin 0 ° )
¿𝟐¿2 (1+𝑖 ∙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.
𝑧𝟏=3√8 [𝑐𝑜𝑠( 0 °+2𝜋 ° ∙𝟏4 )+𝑖𝑠𝑖𝑛( 0 °+2𝜋 ° ∙𝟏4 )]
¿2(cos 2𝜋3 +𝑖 sin 2𝜋3 )
¿−𝟏+𝒊√𝟑¿2(− 12+ 𝑖∙ √32 )
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.
𝑧𝟐=3√8 [𝑐𝑜𝑠( 0 °+2𝜋 ° ∙𝟐4 )+𝑖𝑠𝑖𝑛( 0 °+2𝜋 ° ∙𝟐4 )]
¿2(cos 4 𝜋3 +𝑖 sin 4𝜋3 )
¿−𝟏− 𝒊√𝟑
¿2(− 12+ 𝑖∙(− √32 ))
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
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