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