Mathematics lesson for Form U6 Module: PLANE GEOMETRY …

Mathematics lesson for Form U6

Module: PLANE GEOMETRY AND SOLID FIGURES

Chapter: Complex Numbers

Lesson Title: Polar form (trigonometric form) and

exponential forms of a non-zero complex number

Duration: 120mins

The African Institute for Mathematical Sciences 1

Module 4 : PLANE GEOMETRY AND SOLID

FIGURES

Topic: Complex Numbers

Lesson: Polar form (trigonometric form) and exponential forms of a non-zero complex number


Motivation

As said in the first lesson, Complex numbers have many applications in different fields, including applications in Engineering, Electromagnetism, Quantum Physics and Chaos theory. In Mathematics itself, Complex numbers permits us to solve any polynomial equations.


At the end of this lesson you should be able to: Write or express a non-zero Complex Number in

Polar Form;

Write or express a non-zero Complex Number in Exponential form.

Objectives of Lesson

4The African Institute for Mathematical Sciences

For this lesson you need the following:

A ruler

A pencil

A Pen

Graph paper

A4 papers or your exercise book

NB: For each activity, you are expected to do it on your own before looking at the answer.

Material needed for the lesson


Verification of Pre-requisite

a) Calculate the norm of vectors

𝑢 = 2𝑖 + 2𝑗 = 22, Ԧ𝑣 = 0

3given in the Cartesian

plane.

b) Determine the measure of the angle between 𝑢 𝑎𝑛𝑑 Ԧ𝑣.


Recall, the Norm of a vector is simple the length of the vector.

Generally given any vector 𝑂𝐴 = 𝑎1𝑎2

, then the norm 𝑂𝐴 = 𝑎12 + 𝑎2

2.

1. 𝑢 = 22, Ԧ𝑣 = 0

3

𝑢 = 22 + 22 = 2 2

Ԧ𝑣 = 02 + 32 = 3

2. If 𝜃 is angle between vectors 𝑢 𝑎𝑛𝑑 Ԧ𝑣, then 𝑐𝑜𝑠𝜃 =𝑢.𝑣

𝑢 𝑣

𝑢 = 2 2 and Ԧ𝑣 = 3

𝑢. Ԧ𝑣 = 2 × 0 + 2 × 3 = 0 + 6 = 6

∴ 𝑐𝑜𝑠𝜃 =6

2 2 × 3=

2

2 2=

2

2

𝜃 = 𝑐𝑜𝑠−12

2= 45𝑜


Modulus and Argument of a Complex Number

As earlier seen in the introductory lesson, the point P(x, y) in the plane of coordinates with axes Ox and Oy represents the complex number x + iy

and the number is uniquely represented by that point.


If z = x + iy is a complex number; The Modulus of z is defined as the distance from the origin O to the point Prepresenting z and is denoted as 𝒛 . Thus 𝒛 = 𝑶𝑷 = 𝒓.

Modulus or Magnitude of a Complex Number

Given any complex number z = a + bi, its modulus is 𝑧 and is given by:

𝒛 = 𝒂𝟐 + 𝒃𝟐

Examples: Find the Magnitude of each of the Complex Numbers:

a) 𝑧1 = 4 − 2𝑖; b) 𝑧2 = 7 − 3𝑖; c) 𝑧3 = 3 + 4𝑖

Solutions:

a) 𝑧1 = 4 − 2𝑖 ⟹ 𝑧1 = 42 +−22= 16 + 4 = 20 = 2 5.

b) 𝑧2 = 7 − 3𝑖 ⟹ 𝑧2 = 72 +−32= 58

c) 𝑧3 = 3 + 4𝑖 ⟹ 𝑧3 = 32 + 42 = 25 = 5


Properties of Modulus

Let 𝑧1 and 𝑧2 be two non zero complex numbers with respective magnitudes 𝑧1 and 𝑧2 . Then

i) 𝑧1𝑧2 = 𝑧1 𝑧2 iii) 𝑧𝑛 = 𝑧 𝑛

ii)𝑧1

𝑧2=

𝑧1

𝑧2

Examples:

Let 𝑧1 = 4 − 2𝑖 and 𝑧2 = 2 − 3𝑖 then

𝑧1𝑧2 = 4 − 2𝑖 2 − 3𝑖 = 2 – 16i

𝑧1𝑧2 = 2 − 16𝑖 = 22 + 162 = 260

𝑧1 = 42 +−22 = 20 = 2 5 and 𝑧2 = 22 +−32 = 13

∴ 𝑧1 𝑧2 = 20 × 13 = 260 = 𝑧1𝑧2


Similarly:

Let 𝑧1 = 4 − 2𝑖 and 𝑧2 = 2 − 𝑖 then

4 −2𝑖

2−𝑖=

4−2𝑖 2+𝑖

2−𝑖 2+𝑖= 2 ⟹

𝑧1

𝑧2= 2 = 2

𝑧1

𝑧2=

4−2𝑖

2−𝑖=

2 5

5= 2

∴4−2𝑖

2−𝑖= 2


Argument of a non zero complex number

Representation of the complex number

z = x + yi .

Any complex number z = x + yi can be represented by a point P as on an Argand diagram by the side. While the length of the line segment from the origin to the point gives the magnitude or modulus r of the complex number, the angle 𝜽, measured from the positive real axis to the line segment is called the Argument of the complex number and is denoted arg(z).

The numeric value is given by the angle in radians and is positive if measured counterclockwise.


Calculating Argument of z

Given any complex number z = a + bi, its argument is the angle it makes with the positive real axis within the interval −𝜋, 𝜋 and is calculated using

𝑎𝑟𝑔 𝑧 = 𝑎𝑟𝑐𝑡𝑎𝑛𝑏

𝑎.

Let 𝑎𝑟𝑐𝑡𝑎𝑛𝑏

𝑎= 𝜃

For a complex number z in the first quadrant, 𝒂𝒓𝒈 𝒛 = 𝜽

In the second quadrant the angle is got by subtracting 𝜃, from 𝜋. That is 𝒂𝒓𝒈 𝒛 = 𝝅 - 𝜽

When the complex number falls in the 3rd quadrant 𝒂𝒓𝒈 𝒛 = –(𝝅 - 𝜽) = 𝜽- 𝝅 .

When the complex number is in the 4th quadrant, 𝒂𝒓𝒈 𝒛 = – 𝜽

These are the Principal values of the argument of the complex number.


The different Quadrants .


Consider the complex numbers:

𝑧1 = 1 + 𝑖; 𝑧2 = 2 + 2 3 𝑖; 𝑧3 = −2 + 2 3 𝑖 ; 𝑧4 = −1 − 1 3 𝑖

Represent each on an Argand diagram and determine the argument of each.

1) 𝑧1 = 1 + 𝑖; 𝑧1 is in the 1st quadrant

𝑎𝑟𝑔 𝑧1 = 𝑎𝑟𝑐𝑡𝑎𝑛1

1⟹ 𝜃 =

𝜋

4⟹ 𝑎𝑟𝑔 𝑧 =

𝜋

4


2. 𝑧2 = 2 + 2 3 𝑖; 𝑧2 is in the 1st quadrant

𝑎𝑟𝑔 𝑧2 = 𝑎𝑟𝑐𝑡𝑎𝑛2 3

2= 3 ⟹ 𝜃 =

𝜋

3


𝑧3 = −2 + 2 3 𝑖

𝑎𝑟𝑔 𝑧3 = 𝑎𝑟𝑐𝑡𝑎𝑛2 3

2

= 3 ⟹ 𝛼 =𝜋

3𝛼 is the angle in red.

While 𝜃 𝑖𝑠 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒 𝑖𝑛 𝑏𝑙𝑢𝑒

𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒 𝑧3makes with the positive real axis.

Since 𝑧3 is in the 2nd

quadrant so

𝑎𝑟𝑔 𝑧3 = 𝜋 -𝜋

3= 2𝜋

3


Consider𝑧4 = −1 − 1 3 𝑖

tan𝛼 =1 3

1⟹ 𝛼 = 3

𝑎𝑟𝑐𝑡𝑎𝑛 3 ⟹ 𝛼

𝛼 =𝜋

3

𝑎𝑟𝑔 𝑧4 =𝜋 +𝜋

3=

4𝜋

3out of

the range, so we measure in clockwise direction, hence

𝑎𝑟𝑔 𝑧4 = -(𝜋 −𝜋

3) = -

2𝜋

3


𝑧5 = 2 − 2𝑖

𝑡𝑎𝑛𝜃 =−2

2⟹ 𝜃 =

𝑎𝑐𝑟𝑡𝑎𝑛 −1 = −𝜋

4

arg(𝑧5 ) = −𝜋

4

𝑧6 = −2𝑖

Thus 𝑎𝑟𝑔 𝑧6 =3𝜋

2.

But 3𝜋

2is out of the

domain

So 𝑎𝑟𝑔 𝑧6 = -𝜋

2


An argument is not unique. If 𝜃 𝑖𝑠 𝑎𝑛 𝑎𝑟𝑔 𝑧 then 𝜃 + 2𝑘𝜋 is another argument for 𝑧.

Two complex numbers are equal iff their moduli are equal and their argument differ by integral multiples of 𝟐𝝅 𝒓𝒂𝒅𝒊𝒂𝒏𝒔.

Special Cases:

In some cases the complex number is either purely imaginary or purely real

1. If z is purely imaginary, then z = iy. Its argument is given by:

arg(z) = ቐ

𝜋

2𝑖𝑓 𝑦 > 0 ; 𝑒𝑥𝑎𝑚𝑝𝑙𝑦 𝑧 = 3𝑖

−𝜋

2𝑖𝑓 𝑦 < 0; 𝑒𝑥𝑎𝑚𝑝𝑙𝑒 𝑧 = −3𝑖

2. If z is purely real then z = x. Its argument is given by:

arg(z) =ቊ0 𝑖𝑓 𝑥 > 0; 𝑒𝑥𝑎𝑚𝑝𝑙𝑒 𝑧 = 5𝜋 𝑖𝑓 𝑥 < 0; 𝑒𝑥𝑎𝑚𝑝𝑙𝑒 𝑧 = −5



https://qph.fs.quoracdn.net/main-qimg-12f6bf681a0bb3a3529eccd0fd4c2a08

Properties of arguments

Let z and w be two complex numbers then:1. a𝑟𝑔 𝑧𝑤 = arg(𝑧) + arg(𝑤)

2. arg ҧ𝑧 = 𝑎𝑟𝑔1

𝑧= −𝑎𝑟𝑔𝑧

This implies the argument of the product is the sum of the respective arguments.Examples: Given z = 3 + 3i and w = -3 + 3i 𝑧𝑤 = 3 + 3𝑖 −3 + 3𝑖 = −18 ∴ 𝑎𝑟𝑔 𝑧𝑤 = 𝜋

Similarly, arg(𝑧) =𝜋

4𝑎𝑛𝑑 arg(𝑤) = 𝜋 −

𝜋

4=

3𝜋

4∴ 𝑎𝑟𝑔 𝑧𝑤 =

𝜋

4+

3𝜋

4= 𝜋

2. arg𝑧

𝑤= arg 𝑧 − arg 𝑤

The argument of a quotient of Complex numbers is equal to the difference of their arguments.Examples

Given z = 3 + 3i and w = -3 + 3i. Find arg𝑧

𝑤. arg(𝑧) =

𝜋

4and arg(𝑤) =

3𝜋

4.

∴ 𝑎𝑟𝑔(𝑧

𝑤) =

𝜋

4-3𝜋

4= −

𝜋

2


Properties of arguments Cont.

3. 𝑎𝑟𝑔 ҧ𝑧 = −𝑎𝑟𝑔𝑧 ;

Example:

Let z = 3 + 3i arg 𝑧 =𝜋

4; ҧ𝑧 = 3 − 3𝑖 ∴ 𝑎𝑟𝑔 ҧ𝑧 = −

𝜋

4

4. 𝑎𝑟𝑔𝑧𝑛 = 𝑛𝑎𝑟𝑔𝑧, 𝑛 ∈ ℕ

Example:

Let z = 3 + 3i; find 𝑎𝑟𝑔𝑧5

arg 𝑧 =𝜋

4⟹ 𝑎𝑟𝑔𝑧5 = 5 ×

𝜋

4=

5𝜋

4


Polar or Trigonometric Form of a non zero complex number

Let 𝑧 = 𝑎 + 𝑏𝑖 , modulus of z is 𝑧 = 𝑟 and argument of z is 𝑎𝑟𝑔 𝑧 = 𝛼.

Then 𝑐𝑜𝑠𝛼 =𝑎

𝑟⟹ 𝑎 = 𝑟𝑐𝑜𝑠𝛼

and, 𝑠𝑖𝑛𝛼 =𝑏

𝑟⟹ 𝑏 = 𝑟𝑠𝑖𝑛𝛼

Substituting these values of a and bin z will give:𝑧 = 𝑟𝑐𝑜𝑠𝛼 + i𝑟𝑠𝑖𝑛𝛼 = 𝑟(𝑐𝑜𝑠𝛼 +

In the diagram above, 𝑎 = 𝑟𝑐𝑜𝑠𝛼𝑎𝑛𝑑 𝑏 = 𝑟𝑠𝑖𝑛𝛼. P is the point representing z = a + bi, it follows that z can be written in the form:𝑟𝑐𝑜𝑠𝛼 + 𝑖𝑠𝑖𝑛𝛼


Exponential Form of a non zero complex numberEuler’s Formula

Consider the non zero complex number given in polar form as:

z = 𝑟; 𝛼

We can express this same complex number as 𝒛 = 𝒓𝒆𝒊𝜶 . This is the Exponential form of the complex number z.

We can say

𝒛 = 𝒂 + 𝒃𝒊 = 𝒓 𝒄𝒐𝒔𝜶 + 𝒊𝒔𝒊𝒏𝜶 = 𝒓; 𝜶 = 𝒓𝒆𝒊𝜶

Euler's formula, named after Leonhard Euler, is a Mathematics formula in complex analysis that establishes the fundamental relationship between trigonometric functions and the complex exponential function. Euler's formula states that for any real number x:

𝒆𝒊𝒙 = 𝒄𝒐𝒔𝒙 + 𝒊𝒔𝒊𝒏𝒙


where e, is the base of the natural logarithm, iis the imaginary unit, and cos and sin are the trigonometric function cosine and sine respectively, with the argument x given in radians. This complex exponential function is sometimes denoted cisx

When x = 𝜋, the Euler's formula evaluates to:

𝑒𝑖𝜋 + 1 = 0 , which is known as Euler’s Identity.


Leonhard Euler


Geometrical representation of Polar Form and Exponential Form


Remark:

If 𝑧 = 𝑎 + 𝑏𝑖 then 𝑧 = 𝑎2 + 𝑏2

𝑐𝑜𝑠𝛼 =𝑎

𝑎2 + 𝑏2𝑎𝑛𝑑 𝑠𝑖𝑛𝛼 =

𝑏

𝑎2 + 𝑏2

∴ 𝑧 = 𝑎2 + 𝑏2𝑎

𝑎2 + 𝑏2+ 𝑖

𝑏

𝑎2 + 𝑏2


Example 1Express the complex number in polar form and also in exponential form.

1. 𝑧1 = 1 + 𝑖 3

Solution:

𝑧1 = 1 + 𝑖 3⟹ 𝑧1 = 2 ∴ 𝑟 = 2 ⟹ 𝑧1 = 21

2+ 𝑖

3

2

𝑐𝑜𝑠𝛼 =1

2𝑎𝑛𝑑 𝑠𝑖𝑛𝛼 =

3

2⟹𝛼 =

𝜋

3

𝑧1= 2;𝜋

3in Polar or Trigonometric Form and

𝑧1=2𝑒𝑖𝜋

3 in Exponential Form


Example 2

Express the complex number 𝑧2 = 1 + 𝑖 in polar form and also in exponential form.

2) 𝑧2 = 1 + 𝑖 ⟹ 𝑧2 = 2,⟹ 𝑧2 = 21

2+ 𝑖

1

2

𝑐𝑜𝑠𝛼 =2

2𝑎𝑛𝑑 𝑠𝑖𝑛𝛼 =

2

2⟹𝛼 =

𝜋

4

𝑧2 = 2;𝜋

4in Polar form

𝑧2 = 2𝑒𝑖𝜋

4

∴ ) 𝑧2 = 1 + 𝑖 = 2;𝜋

4= 2𝑒𝑖

𝜋

4


Example 3a) Express 8 𝑐𝑜𝑠

7𝜋

6+ 𝑖𝑠𝑖𝑛

7𝜋

6in rectangular form

Modulus r = 8.

𝑥 = 𝑟 cos 𝜃 ⟹ 𝑥 = 8 𝑐𝑜𝑠7𝜋

6= −8𝑐𝑜𝑠

𝜋

6= −8

3

2= −4 3

𝑦 = 𝑟𝑠𝑖𝑛𝜃 ⟹ 𝑦 = 8𝑠𝑖𝑛7𝜋

6= 8𝑠𝑖𝑛

7𝜋

6= −8𝑠𝑖𝑛

𝜋

6= −8

1

2= −4

x = −4 3 and y = -4

∴ 8 𝑐𝑜𝑠7𝜋

6+ 𝑖𝑠𝑖𝑛

7𝜋

6= −4 3 - 4i


Example 4

𝑧1 = 2 𝑐𝑜𝑠𝜋

6+ 𝑖𝑠𝑖𝑛

𝜋

6; 𝑎𝑛𝑑 𝑧2 = 𝑐𝑜𝑠

𝜋

3+ 𝑖𝑠𝑖𝑛

𝜋

3Express 𝑧1 ÷ 𝑧2 𝑖𝑛 𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑓𝑜𝑟𝑚.

Solution:

𝑧1 = 2; 𝑧2 = 1; and𝑧1𝑧2

=2

1= 1

Let 𝜃 𝑏𝑒 𝑡ℎ𝑒 𝑎𝑟𝑔𝑢𝑚𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑒𝑠𝑢𝑙𝑡𝑖𝑛𝑔 𝑐𝑜𝑚𝑝𝑙𝑒𝑥 𝑛𝑢𝑚𝑏𝑒𝑟 𝑡ℎ𝑒𝑛:

𝜃 = 𝜃1 − 𝜃2 =𝜋

6−𝜋

3= −

𝜋

6

∴ 2 𝑐𝑜𝑠𝜋

6+ 𝑖𝑠𝑖𝑛

𝜋

6÷ 𝑐𝑜𝑠

𝜋

3+ 𝑖𝑠𝑖𝑛

𝜋

3= 2 𝑐𝑜𝑠 −

𝜋

6+ 𝑖𝑠𝑖𝑛 −

𝜋

6

= 23

2+ −

1

2𝑖 = 3 − 𝑖


Other examples

1) 𝑧 = −2𝑖; 𝑧 = 2; 𝑎𝑟𝑔 𝑧 = −𝜋

2⟹ 𝑧 = 2; −

𝜋

2in polar form

and 𝑧 = 2𝑒𝑖−𝜋

2 in Exponential form.

2) 𝑧 = 2 − 2𝑖; 𝑧 = 2 2 ; 𝑎𝑟𝑔 𝑧 =7𝜋

4= −

𝜋

4⟹ 𝑧 = 2 2; −

𝜋

4

and 𝑧 = 2 2 𝑒𝑖−𝜋

4 is the Exponential form.

3) 𝑧 = −1 − 𝑖 3; 𝑧 = 2 ; 𝑎𝑟𝑔 𝑧 =4𝜋

3

and 𝑧 = 2𝑒𝑖4𝜋

3 is the Exponential form


Let 𝑧 = 𝑟, 𝛼 = 𝑟𝑒𝑖𝛼 and 𝑧′ = 𝑟′; 𝛼′ = 𝑟′𝑒𝑖𝛼′ be two complex numbers then:


Polar Form Exponential Form

ҧ𝑧 = 𝑟; −𝛼 ҧ𝑧 = 𝑟𝑒−𝑖𝛼

−𝑧 = 𝑟; −𝛼 −𝑧 = 𝑟𝑒𝑖 𝛼+𝜋

𝑧 × 𝑧′ = 𝑟𝑟′; 𝛼 + 𝛼′ 𝑧 × 𝑧′ = 𝑟𝑟′𝑒𝑖 𝛼+𝛼′

1

𝑧=

1

𝑟;−𝛼

1

𝑧=1

𝑟𝑒−𝑖𝛼

𝑧

𝑧′=

𝑟

𝑟′; 𝛼 − 𝛼′

𝑧

𝑧′=

𝑟

𝑟′𝑒𝑖 𝛼−𝛼′

Home work1. Express each complex number in polar form: i) -2 + I ; ii) 2 3 − 3𝑖.

2. Express the following complex numbers in rectangular form:

i) 2 𝑐𝑜𝑠5𝜋

4+ 𝑖𝑠𝑖𝑛

5𝜋

4; ii) 2𝑖𝑠𝑖𝑛 −

𝜋

2+ 2𝑐𝑜𝑠 −

𝜋

2

ii) 2 𝑐𝑜𝑠𝜋

6+ 𝑖𝑠𝑖𝑛

𝜋

6

3. Find the zeros of 𝑓 𝑥 = 𝑥2 − 4𝑥 + 8. Graph them on the complex plane.

4. Determine the loci of the point M(z) such that a) 𝑧 − 1 + 𝑖 = 1 + 3𝑖 ; b) 𝑧 + 2 − 3𝑖 = 𝑧 + 1 + 𝑖

5. Given that 𝑧1 =6−𝑖 2

2and 𝑧2 = 1 − 𝑖.

a) Find the modulus and argument of each complex number.

b) Write the complex numbers in polar form and in exponential form.


Next Lesson will be on powers and roots of a non zero complex number


References

1. Advanced Level Pure Mathematics Made Easy, Ewane Roland Alunge, First Edition 2017

2. Advanced Mathematical Concepts, Glencoe McGraw-Hill;

3. Pure Mathematics with Mechanics Teaching Syllabus Second Cycle Secondary General Education, 2019 Cameroon

4. https://qph.fs.quoracdn.net/main-qimg-12f6bf681a0bb3a3529eccd0fd4c2a08

5. https://en.wikipedia.org/wiki/Euler%27s_formula

6. https://www.mathsisfun.com/algebra/eulers-formula.html


https://qph.fs.quoracdn.net/main-qimg-12f6bf681a0bb3a3529eccd0fd4c2a08

https://en.wikipedia.org/wiki/Euler's_formula

https://www.mathsisfun.com/algebra/eulers-formula.html

THANKS VERY MUCH AND GOOD LUCK IN YOUR STUDIES


Date post:	27-Jan-2022
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Mathematics lesson for Form U6 Module: PLANE GEOMETRY …

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