Section 3. De Moivre s theorem

3.0 De Moivre’s theorem for rational indices

Theorem 3.0: (De Moivre’s theorem for integers) If n∈ Z , then (cosθ+isin )θ ⁿ =cosnθ+isinnθ.

e.g.3.0 Convert 2

3 1+ i

and 2

3 1− i

to their polar forms. Hence find the value of

2007

2007 )

2 3 (1 2 )

3

(1+ i + + i

.

e.g.3.1 Let n∈ N.

a) i) Find (1+ i)ⁿ.

ii) Evaluate

C C C C

C C C C

∑

k k n k

n

C

=

∑

1 k

k n k

n

C

e.g.3.2 Express ¹ 1

+ +

+ −

⎛

⎝⎜ ⎞

⎠⎟ sin cos sin cos

θ θ

θ θ

i i

n

in polar form and evaluate (1 sin cos ) ( sin cos )

6 6 1

6 6

6 6

+ π+ π − + π − π

i i .

e.g.3.3 a) Show that if x+y+z =0, then x³ +y³+z³ =3xyz.

b) Hence show that if cosα +cosβ +cosγ =sinα+sinβ +sinγ =0, then )

cos(

3 3 cos 3

cos

cos3α+ β + γ = α+β +γ and )

sin(

3 3 sin 3 sin

sin3α + β + γ = α +β+γ .

Remark:

Besides cisθ, we can use e^iθ to denote cosθ+ isinθ, which is called the Euler’s formula. Then De Moivre’s theorem can be written as (e^iθ)ⁿ =e^inθ.

3.1 Applications to trigonometric identities

Direct applications of De Moivre’s theorem and the binomial theorem, we are able to express

i) multiple angles such as sin nθ and cosnθ in terms of sinθ and cosθ , and ii) powers of sinθ and cosθ come back again into multiple angles.

The following examples is to illustrate the method.

e.g.3.4 Expand (cosθ +isinθ)³ and hence find cos3θ and sin3θ in terms of θ

sin and cosθ and tan3θ in terms of tanθ.

e.g.3.5 Express the following: i) cos5θ in terms of cosθ; ii) sin 5θ in terms of sinθ; iii) tan 5θ in terms of tanθ.

Remark:

The following formulas are useful:

Let z=cosθ+isinθ , then z + =1z

2 cosθ , z

z i

− =1

2 sinθ and

zⁿ +z1_n = n 2 cos θ, _z

z i n

n

− 1n =

2 s in θ .

e.g.3.6 Show that i) (cos4 4cos 3) 8

cos⁴θ =1 θ + 2θ + and

ii) (sin5 5sin3 10sin ) 16

sin⁵θ = 1 θ − θ + θ .

e.g.3.7 Express cos⁵θsin³θ in terms of multiple angles.

e.g.3.8 i) Express cos 6θ in terms of cosθ.

ii) Solve the equation a) 32x³ −48x² +18x−1=0 and b) 64x³ −96x² +36x− =3 0.

iii) Show that sec² sec² sec² 18

5 18

7 18 12

π π π

+ + = .

e.g.3.9 Evaluate the series i) cosk

k

n θ

∑

and ii) _kⁿ

k n

C

∑

.

Past Paper:

98-CE-I-4; 2000-I-6

3.2 n th roots of a complex number

Definition 3.0 A complex number ω is called a n th root of another complex number z if ωⁿ = . z

Remark: For any complex number z₀, the equation ωⁿ =z₀ has n roots, hence there are n n th roots of z₀.

Theorem 3.1

Let z =rcisθ , where r > 0 , be any complex number, then z has exactly n distinct n th roots

z r cis k

k n

= n¹ +2 (θ π)

, where k = 0 1 2 3, , , ,L,n−1.

Remark: The n th roots of z can be written as z r cis k

k = n¹ +n2

(θ π)

, where k =m m, +1,m+2,m+3,L,m n+ −1 for any integer m.

e.g.3.10 Find the fourth roots of 1+i. Locate the four roots on the Argand plane.

Definition 3.1 A complex number ω is called a nth root of unity if ωⁿ = 1.

e.g.3.11 Solve the equation z³ =1 and represent the solutions on the Argand diagram.

If ω is a complex root of above equation, show that 1+ω+ω² =0.

Remark: From the above theorem, when r =1,θ = 0the nth roots of unity are

n cis n cis n

cis n

cis nπ π π 2( 1)π

,..., , 6

, 4 , 2

1 −

.

If we write ω _{= cis} π n

2 , the nth roots of unity can be expressed as

1, ,ω ω ω², ³,_L,ωⁿ⁻¹.

The points on the Argand plane representing the roots 1, ,ω ω ω², ³,_L,ωⁿ⁻¹ are vertices of a regular n-sided polygon inscribed in the unit circle.

e.g.3.12 Let n∈ N , n>1 and ω π

= cis n

2 , find the value of

i) 1+ +ω ω² +ω³+ +_L ωⁿ⁻¹

iii) (1−ω)(1−ω )(1−ω )_L(1−ω )

iv) 1+ω^m+ω^2m+ω^3m+ +_L ω^(n−1)m where m∈ Z .

Remark: In some cases, the fact that

“ωⁿ = 1 and ω ≠ 1 ⇒ + +1 ω ω²+ω³+ +_L ωⁿ⁻¹ = 0” is useful in calculation

e.g.3.13 Let ω is a complex cube root of unity. Show that i) a³−b³ = (a−b a)( −bω)(a−bω²)

ii) a³+b³ = (a+b a)( +bω)(a+bω²)

iii) a² +b²+c²−ab−bc−ca =(a+bω+cω²)(a+bω²+cω) iv) a³+b³+c³−3abc= (a+ +b c a)( +bω+cω²)(a+bω² +cω).

e.g.3.14 By considering the expansion of (1+ x)ⁿ , with x= 1, ,ω ω² , where

ω π

= cis2

3 . Evaluate i)

C C C

C C C

e.g.3.15 Let ε π

= cis n

2 , n is a natural number and n > 1..

i) Find 1+ +ε ε² + +ε³ ...+εⁿ⁻¹. ii) Evaluate cos2

1

1 k

k n

n π

=

∑

1 k

k n

n π

=

∑

iii) Show that 1 2 3

1

2 1

+ + + + = −

− ε ε ε −

... n _n nε .

iv) Evaluate k k

k n

n

cos2

1

1 π

=

∑

k n

sin2

1

1 π

=

∑

Past Paper:

98-CE-I-12; 99-CE-I-5 3.3 Solution of Equations

e.g.3.16 Solve the equation z³ =8(1−z)³.

e.g.3.17 Solve the equation (z+1)ⁿ = (z−1) ,ⁿ n∈N and n >1.

e.g.3.18 i) Solve the equation (1+z)²ⁿ⁺¹= −(1 z)²ⁿ⁺¹. ii) Hence show that tanπ tan π

5 2

5 5

⋅ = .

Class Practices:

3.1 Show that the roots of the equation (z+1)²ⁿ +(z−1)²ⁿ =0, where n is a positive integer, are purely imaginary.

Past Paper:

2001-AL-I-11; 2003-AL-I-12(a)(i)

Section 4. Complex-valued functions

Definition 4.0

A function whose range is the set C is called a complex-valued function.

For instance,i) f:[ ,0 2π)→ C, f ( )θ =cosθ+isinθ.

ii) g i g z z z i :C\ { }→C, ( )= +

−

3 4

.

e.g.4.0 Let a, b be real constants. The function f :R →C is defined by

f t a b it

( ) ( it )

= + +

− 1

1 .

Show that the image of R under f lies in a circle.

e.g.4.1 Let f :C\ { }0 → be defined by C f z z ( )= + z1

.

i) Let ω = rcisθ, where r > 0, express f ( )ω in the form a + bi, where a b, ∈ R .

ii) Find and sketch the image of each of the following set under f:

a) S₁ ={z∈C:z =1}

iii) Is f injective?

e.g.4.2 Let f :C\ { }− →1 C\ { }1 be defined by f z z ( )= z

1+ . i) What is the image of the imaginary axis under f?

ii) Prove that f is bijective.

e.g.4.3 Let a b, ∈C with a =1,b <1, D ={z∈C:z ≤1}.

The function f D: → C is defined by f z a z b ( ) ( zb)

= −

1− . Show that i) f z( ) =1 ∀ =z 1; ii) f z( ) <1 ∀ <z 1.

e.g 4.4 A mapping f :C→ is said to be real linear if C

f (αz₁+βz₂)=αf z( )₁ +βf z( )₂ ∀α β, ∈R,z z₁, ₂ ∈C.

Let φ:C→ be a real linear mapping. C

If φ is non-constant such that φ(z z_{1 2})=φ( ) ( )z₁ φ z₂ ∀z z₁, ₂ ∈C, show that i) φ( )1 =1 and hence φ( )x = x ∀ ∈R . x

Past Paper:

89-AL-I-12; 90-AL-I-10; 91-AL-I-9; 94-AL-I-11