Complex Number Part 2 Learning Outcome: Understand the graphical representation of complex numbers through an Argand diagram.

Understand complex numbers in other forms.

~mys2020~

Complex Number (Graphical Representation) 𝑰𝒎(𝒛)

Argand Diagram

𝒁 = 𝒂 + 𝒃𝒊

𝒃

Modulus 𝒁 =

𝒂 𝟐 + 𝒃𝟐 Argument

𝜽 = 𝒂𝒓𝒈 𝒛 = tan−𝟏

𝒃 𝒂 𝑹𝒆(𝒛)

𝒂

** 𝜽 = angle between the positive 𝒙 − 𝒂𝒙𝒊𝒔 and the line Z. ~mys2020~

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Complex Number (Graphical Representation) Examples: Represent 𝑧 = 4 + 3𝑖 in an Argand diagram and find its modulus and argument.

𝑰𝒎(𝒛)

Modulus: 𝒁 =

𝒂 𝟐 + 𝒃𝟐

= 𝟒𝟐 + 𝟑𝟐 = 25 =𝟓 𝒁 = 𝟒 + 𝟑𝒊

𝟑

Argument:

𝟓

𝒃 𝒂 𝟑 = tan−𝟏 𝟒 = 𝟑𝟔. 𝟖𝟕°

𝒂𝒓𝒈 𝒛 = tan−𝟏

𝜽 = 𝟑𝟔. 𝟖𝟕° 𝟒

𝑹𝒆(𝒛)

~mys2020~

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Complex Number (Graphical Representation) Exercise 1: Represent 𝑧 = −3 + 4𝑖 in an Argand diagram and find its modulus and argument. 𝑰𝒎(𝒛)

Modulus: 𝒁 =

=𝟓

Argument: 𝒂𝒓𝒈 𝒛 =

𝑹𝒆(𝒛)

= 𝟏𝟐𝟔. 𝟖𝟔𝟗𝟗° ~mys2020~

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Complex Number (Graphical Representation) Exercise 2: Represent 𝑧 = −6 − 5𝑖 in an Argand diagram and find its modulus and argument. Modulus: 𝒁 =

= 𝟔𝟏

Argument: 𝒂𝒓𝒈 𝒛 =

= 𝟐𝟏𝟗. 𝟖𝟎𝟓𝟔° ~mys2020~

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Complex Number (Graphical Representation) Exercise 3: Represent 𝑧 = 4 − 7𝑖 in an Argand diagram and find its modulus and argument.

Modulus: 𝒁 =

= 𝟔𝟓

Argument: 𝒂𝒓𝒈 𝒛 =

= 𝟐𝟗𝟗. 𝟕𝟒𝟒𝟗° ~mys2020~

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Complex Number (Graphical Representation) Exercise 4: Represent 𝑧 = 5 + 9𝑖 in an Argand diagram and find its modulus and argument.

Modulus: 𝒁 =

= 𝟏𝟎𝟔

Argument: 𝒂𝒓𝒈 𝒛 =

= 𝟔𝟎. 𝟗𝟒𝟓𝟒° ~mys2020~

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Complex Number (Graphical Representation) Exercise 5: Represent 𝑧 = −8 + 4𝑖 in an Argand diagram and find its modulus and argument. Modulus: 𝒁 =

= 𝟖𝟎

Argument: 𝒂𝒓𝒈 𝒛 =

= 𝟏𝟓𝟑. 𝟒𝟑𝟒𝟗° ~mys2020~

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Complex Number (Graphical Representation) Exercise 6: Given 𝑧1 = 5 + 3𝑖 and 𝑧2 = 4 − 6𝑖, represent each of the following in an Argand diagram and find its modulus and argument. a.

𝑧1 + 𝑧2

b.

= 𝟗𝟎 & 𝟐𝟏𝟗. 𝟖𝟎𝟓𝟔° ~mys2020~

𝑧1 − 𝑧2

= 𝟖𝟐 & 𝟖𝟑. 𝟔𝟓𝟗𝟖° 9

Complex Number (Graphical Representation)

c.

𝑧1 𝑧2

d.

= 𝟏𝟕𝟔𝟖 & 𝟑𝟑𝟒. 𝟔𝟓𝟑𝟖° ~mys2020~

𝑧1 𝑧2

= 𝟎. 𝟖𝟎𝟖𝟔 & 𝟖𝟕. 𝟐𝟕𝟑𝟕° 10

Complex Number (Polar form, Exponential form and Trigonometric form)

Polar form

Cartesian form

𝑧 = 𝑧 ∠𝜃° @ 𝑧 = 𝑟∠𝜃°

𝑧 = 3 + 5𝑖

Trigonometric form

Exponential form

𝑧 = |𝑧| cos 𝜃 + 𝑖 sin 𝜃 @ 𝑧 = 𝑟 cos 𝜃 + 𝑖 sin 𝜃

𝑧 = 𝑧 𝑒 𝜃𝑖 𝜃 𝑖𝑛 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 @ 𝑧 = 𝑟𝑒 𝜃𝑖

~mys2020~

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Complex Number (Polar form, Exponential form and Trigonometric form) Example: Convert z = 3 + 5𝑖 into trigonometric form, polar form and exponential form. Step 1: Find the values of modulus and argument 5 3 = 𝟓𝟗. 𝟎𝟑𝟔𝟐°

𝑟 = 32 + 52 = 𝟑𝟒

𝜃 = tan−1

Step 2: Plot the complex number in an Argand diagram 𝑰𝒎(𝒛) 𝟓

𝒓 = 𝟑 + 𝟓𝒊

𝜽 = 𝟓𝟗. 𝟎𝟑𝟔𝟐° 𝟑

𝑹𝒆(𝒛) ~mys2020~

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Complex Number (Polar form, Exponential form and Trigonometric form) Step 3: Substitute the values of 𝑟 and 𝜃 into the formulas. Trigonometric form

𝑧 = 𝑟 cos 𝜃 + 𝑖 sin 𝜃 = 34 cos 59.0362 + 𝑖 sin 59.0362 Polar form 𝑧 = 𝑟∠𝜃° = 34∠59.0362° Exponential form 𝜃0 = 59.0362 × = 1.0304 𝑟𝑎𝑑

𝜋 180°

𝑧 = 𝑟𝑒 𝜃𝑖 = 34𝑒1.0304𝑖 ~mys2020~

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Complex Number (Polar form, Exponential form and Trigonometric form) Exercise 1: Convert z = 4 + 6𝑖 into trigonometric form, polar form and exponential form.

~mys2020~

= 𝟓𝟐 cos 𝟓𝟔. 𝟑𝟎𝟗𝟗° + 𝒊 sin 𝟓𝟔. 𝟑𝟎𝟗𝟗° = 𝟓𝟐∠𝟓𝟔. 𝟑𝟎𝟗𝟗° = 𝟓𝟐𝒆𝟎.𝟗𝟖𝟐𝟖𝒊 14

Complex Number (Polar form, Exponential form and Trigonometric form) Exercise 2: Convert 𝑧 = 25∠48° into trigonometric form, cartesian form and exponential form.

= 𝟐𝟓 cos 𝟒𝟖° + 𝒊 sin 𝟒𝟖° = 𝟏𝟔. 𝟕𝟐𝟕𝟓 + 𝟏𝟖. 𝟓𝟕𝟕𝟓𝒊 = 𝟐𝟓𝒆𝟎.𝟖𝟑𝟕𝟖𝒊 ~mys2020~

15

Complex Number (Polar form, Exponential form and Trigonometric form) Exercise 3: Given 𝑧1 = 12∠45° , 𝑧2 = 24 cos 10° + 𝑖 sin 10° and 𝑧3 = 14𝑒 0.342𝑖 . Calculate: a.

𝑧1 𝑧2

b.

(in polar form)

= 𝟐𝟖𝟖∠𝟓𝟓°

𝑧1 𝑧3

(in polar form)

= 𝟏𝟔𝟖∠𝟔𝟒. 𝟓𝟗𝟓𝟐° ~mys2020~

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Complex Number (Polar form, Exponential form and Trigonometric form) c.

𝑧2 𝑧1

d.

(in trigonometric form)

= 𝟐 cos −𝟑𝟓 + 𝒊 sin −𝟑𝟓 ~mys2020~

𝑧3 𝑧2

(in exponential form)

= 𝟎. 𝟓𝟖𝟑𝟑𝒆𝟎.𝟏𝟔𝟕𝟓𝒊 17

Complex Number (Polar form, Exponential form and Trigonometric form) Exercise 4: Solve the following expression in an exponential form.

10 cos 200° + 𝑖 sin 200° × 6 cos 10° + 𝑖 sin 10° 20 cos 70° + 𝑖 sin 70

= 𝟑𝒆𝟐.𝟒𝟒𝟑𝟓𝒊 ~mys2020~

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To be continued ~mys2020~

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