• Represent a complex number and its conjugate on the Argand plane and find its modulus and (principal) amplitude.
• Write a complex number in trigonometric and exponential (Euler) form, and multiply and divide in these forms.
• Use De Moivre’s theorem for powers, identities and the n-th roots of a complex number.
• Know the cube roots of unity, their properties and their applications.

🌳 Concept tree

• Lesson 1 Complex Numbers in Trigonometric and Exponential Form and De Moivres Theorem
• ├── 2-1 (a) Polar coordinates and the Argand plane

│ ├── Polar and Cartesian coordinates

│ ├── Argand plane

│ └── Symmetry on the Argand plane

1. ├── 2-1 (b) Modulus
2. amplitude and principal amplitude

│ ├── Definitions

• │ ├── Finding the principal amplitude from the quadrant of $z$

│ └── Properties of the modulus and amplitude

• ├── 2-1 (c) The trigonometric (polar) form of a complex number

│ ├── The form

│ ├── Useful special values

│ └── Numbers that are not in standard form

• ├── 2-1 (d) Product and quotient in trigonometric form

│ ├── Rules

│ └── Consequences

├── 2-1 (e) The exponential (Euler) form

│ ├── Series that lead to Euler's formula

│ ├── Euler's formula

│ ├── Product and quotient in exponential form

1. │ └── Expressing sine and cosine with exponentials
2. ├── 2-2 (a) De Moivre's theorem for a positive integer power

│ ├── Statement

• │ └── Using the theorem to prove trigonometric identities
• ├── 2-2 (b) De Moivre's theorem for a rational power and the roots of a complex number

│ ├── Rational power

│ ├── The n-th roots of a complex number

1. │ ├── Square roots of a complex number in algebraic form
2. │ └── Quadratic equations with complex coefficients

└── 2-3 The cube roots of unity

├── Finding them

├── Properties of the cube roots of unity

└── Applications

Rules

• Let $z_1=r_1(\cos {\theta }_1+i\sin {\theta }_1)$ and $z_2=r_2(\cos {\theta }_2+i\sin {\theta }_2)$.
• Multiplication moduli multiply, amplitudes add. z1z2=r1r2[cos⁡(θ1+θ2)+isin⁡(θ1+θ2)]
• Division moduli divide, amplitudes subtract. z1z2=r1r2[cos⁡(θ1−θ2)+isin⁡(θ1−θ2)]
• Therefore $\mathrm{\mid }{z}_1{z}_2\mathrm{\mid }=\mathrm{\mid }{z}_1\mathrm{\mid }\mathrm{\mid }{z}_2\mathrm{\mid }$, $\mid {\displaystyle \frac{z_1}{z_2}}\mid ={\displaystyle \frac{\mathrm{\mid }{z}_1\mathrm{\mid }}{\mathrm{\mid }{z}_2\mathrm{\mid }}}$, $\arg (z_1{z}_2)={\theta }_1+{\theta }_2$, $\arg {\displaystyle \frac{z_1}{z_2}}={\theta }_1-{\theta }_2$.

✍️ Worked examples

• Example 6 Write $3(\cos {\displaystyle \frac{\pi }{3}}+i\sin {\displaystyle \frac{\pi }{3}})\times 4(\cos {\displaystyle \frac{\pi }{6}}+i\sin {\displaystyle \frac{\pi }{6}})$ in the form $x+yi$.
• Solution:
• Multiply the moduli 3×4=12.
• Add the amplitudes π3+π6=π2.
• The product is $12(\cos {\displaystyle \frac{\pi }{2}}+i\sin {\displaystyle \frac{\pi }{2}})$.
• $=12(0+i)=12i$.
• Example 7 Find ${\displaystyle \frac{6(\cos 100^{\circ }+i\sin 100^{\circ })}{3(\cos 40^{\circ }+i\sin 40^{\circ })}}$ in the form $x+yi$.
• Solution:
• Divide the moduli 6÷3=2.
• Subtract the amplitudes 100∘−40∘=60∘.
• The quotient is $2(\cos 60^{\circ }+i\sin 60^{\circ })$.
• $=2({\displaystyle \frac{1}{2}}+{\displaystyle \frac{\sqrt{3}}{2}}i)=1+\sqrt{3}i$.
• Example 8 Write $1-i$ in trigonometric form, then find $(1-i{)}^8$.
• Solution:
• x=1, y=−1 fourth quadrant, r=2, θ=−π4.
• $1-i=\sqrt{2}[\cos (-{\displaystyle \frac{\pi }{4}})+i\sin (-{\displaystyle \frac{\pi }{4}})]$.
• $(1-i{)}^8=(\sqrt{2}{)}^8[\cos (-2\pi )+i\sin (-2\pi )]$.
• $=16(1+0i)=16$.

Finding them

• Solve z3=1 with De Moivre’s theorem the roots are 1, −12+32i and −12−32i.
• Algebraic check z3−1=(z−1)(z2+z+1)=0 gives z=1 or z=−1±3 i2.
• The square of one complex root equals the other, so the roots are written as $1,\omega ,{\omega }^2$ where:
  $$\omega =-\frac{1}{2}+\frac{\sqrt{3}}{2}i,{\omega }^2=-\frac{1}{2}-\frac{\sqrt{3}}{2}i$$
• On the Argand plane the roots lie on the unit circle, at angles $0^{\circ }$, $120^{\circ }$ and $-120^{\circ }$, and form an equilateral triangle.

Properties of the cube roots of unity

✍️ Worked examples

• Example 19 Find the value of $5+5\omega +5{\omega }^2$ and of $(1-{\displaystyle \frac{2}{\omega }}-{\displaystyle \frac{2}{{\omega }^2}})(3+5\omega +5{\omega }^2)$.
• Solution:
• $5+5\omega +5{\omega }^2=5(1+\omega +{\omega }^2)=5\times 0=0$.
• Use ${\displaystyle \frac{1}{\omega }}={\omega }^2$ and ${\displaystyle \frac{1}{{\omega }^2}}=\omega$.
• First bracket 1−2ω2−2ω=1−2(ω+ω2)=1+2=3.
• Second bracket 3+5(ω+ω2)=3−5=−2.
• The product is $3\times (-2)=-6$.
• Example 20 If $1,\omega ,{\omega }^2$ are the cube roots of unity, prove that $(1-\omega +{\omega }^2)(1+\omega -{\omega }^2)=4$.
• Solution:
• $1+{\omega }^2=-\omega$, so $1-\omega +{\omega }^2=-\omega -\omega =-2\omega$.
• $1+\omega =-{\omega }^2$, so $1+\omega -{\omega }^2=-{\omega }^2-{\omega }^2=-2{\omega }^2$.
• The product is $(-2\omega )(-2{\omega }^2)=4{\omega }^3=4$.
• Example 21 Prove that $x={\displaystyle \frac{-1+\sqrt{3}i}{2}}$ is a root of $x^{10}+x^5+1=0$.
• Solution:
• $x=-{\displaystyle \frac{1}{2}}+{\displaystyle \frac{\sqrt{3}}{2}}i=\omega$, one of the complex cube roots of unity.
• $x^{10}={\omega }^{10}={\omega }^9\cdot \omega =\omega$.
• $x^5={\omega }^5={\omega }^3\cdot {\omega }^2={\omega }^2$.
• $x^{10}+x^5+1=\omega +{\omega }^2+1=0$, so $x$ is a root.
• The same holds for x=ω2 (ω2)10=ω2 and (ω2)5=ω, giving again ω2+ω+1=0.

Applications

• Forming a quadratic equation from two roots that are expressed in $\omega$: find the sum and product of the roots using $1+\omega +{\omega }^2=0$ and ${\omega }^3=1$, then write $x^2-(\text{sum})x+(\text{product})=0$.
• Reducing expressions replace 1/ω by ω2, group terms into 1+ω+ω2 and reduce powers modulo 3.
• Complex numbers are used in many fields, including electricity, dynamics, relativity and other physical applications.