FMA2 Complex numbers Objective Deadlines / Progress

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Express a complex number in a) modulus argument form b) in exponential form

Convert between different forms and solve basic problemsMultiply and divide complex numbers in any form; know how multiplying and dividing affects both the modulus and argument of the resulting complex number

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FMA2 Complex numbers

Modulus argument form of a complex number

Reminder cos(-θ) = cosθ and sin(-θ) = -sinθ

WB1 Express in the modulus-argument form:

a) z = -√3 + i

b) z = 1 – i

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FMA2 Complex numbers

Exponential form of a complex number

If z = x + iy then the complex number can also be written in this way z = reiθ

As before, r is the modulus of the complex number and θ is the argumentThis form is known as the ‘exponential form’

WB2 Express the following complex number in the form reiθ, where -π < θ ≤ π

a) z = 2 – 3i

b) z ¿√2(cos ( π10 )+isin( π10 ))

c) z¿5(cos( π8 )−isin ( π8 ))

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FMA2 Complex numbers WB3

a) Express the following in the form, z = x + iy where x∈ R and y∈R z=√2e

3 π4 i

b) Express the following in the form r(cosθ + isinθ), where x∈ R and y∈R z=2e

23 π5 i

WB4 Use: e iθ=cosθ+isinθ to show that: cosθ=12

(eiθ+e−iθ)

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FMA2 Complex numbers

Multiplication and division

Addition formula for sin and cos

sin (θ1±θ2 )=sin θ1 cosθ2± cosθ1 sinθ2

cos (θ1±θ2 )=cosθ1 cosθ2∓sinθ1sin θ2

In modulus argument form

z1 z2=r 1r2 (cos (θ1+θ2 )+isin (θ1+θ2 )) z1z2

=r1

r2(cos (θ1−θ2 )+isin (θ1−θ2 ))

In exponential form

z1 z2=r 1r2ei (θ1+θ2)

z1z2

=r1

r2e i(θ¿¿ 1−θ2 )¿

WB5 Express the following calculation in the form x + iy:

a¿3(cos 5 π12

+ isin 5 π12 )×4 (cos π

12+ isin π

12 )

b¿2(cos π15

+isin π15 )×3(cos 2π

5−isin 2π

5 )

c)√2(cos π

12+isin π

12 )2(cos 5π

6+isin 5 π

6 )

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FMA2 Complex numbers

WB6 Express the following calculations in the form x + iy:

a¿2eπ i6 ×√3e

πi3

b¿2e

π i3

√3 eπ i6

WB7

Express 2(cos π

12+isin π

12 )√2(cos 5 π

6+isin 5 π

6 ) in the form r e iθ

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FMA2 Complex numbers

WB8 z=2+2 i ,ℑ (zw )=0∧|zw|=3|z|use geometrical reasoning to find the two possibilities for w, giving them in exponential form

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FMA2 Complex numbers

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FMA2 Complex numbers

De Moivres theorem

zn=[r (cosθ+isinθ)]n=r n (cosnθ+isinnθ )

zn=[ ℜiθ ]n=rn e¿ θ

WB9 Let: z = r(cosθ + isinθ) find z2 , z3 , z4…what do you notice ?

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FMA2 Complex numbers WB10

Simplify the following: (cos 9 π

17+isin 9 π

17 )5

(cos 2π17

−isin 2π17 )

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WB11 Express the following in the form x + iy where x Є R and y Є R(1+√3 i)7

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FMA2 Complex numbers

Apply De Moivres theorem to Trigonometric identities

(a+b )n=¿ an+nC1an−1b+nC2a

n−2b2+nC3an−3b3+…………+bn

WB12 Express cos3θ using powers of cosθ

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FMA2 Complex numbers

WB13 Express the following as powers of cosθ: sin 6θsinθ

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FMA2 Complex numbers

Let: z=cosθ+isinθ

Then 1z= (cosθ+isinθ   )−1=cos (−θ)+ isin(−θ)cosθ−isinθ

From which we can get these resultsz+1z=2cosθ

z−1z=2isinθ

zn+ 1zn

=2 cos (nθ)

zn− 1zn

=2isin(nθ)

WB14 Express cos5θ in the form acos5θ + bcos3θ + ccosθ Where a, b and c are constants to be found

WB15 Show that: sin3θ=−14

sin3θ+ 34sinθ

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FMA2 Complex numbers

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FMA2 Complex numbers WB16

a) Express sin4θ in the form: dcos 4θ+ecos2θ+ f Where d, e and f are constants to be found.

b) Hence, find the exact value of the following integral: ∫0

π2

sin 4θdθ

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FMA2 Complex numbers

WB17 a) Express sin4θ in the form: dcos 4θ+ecos2θ+ f Where d, e and f are constants to be found.

b) Hence, find the exact value of the following integral: ∫0

π2

sin 4θdθ

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FMA2 Complex numbers

You already know how to find real roots of a number, but now we need to find both real roots and imaginary roots! We need to apply the following results

1) If: z=r (cosθ+isinθ )

Then: z=r (cos (θ+2kπ )+isin (θ+2kπ )) where k is an integer

This is because we can add multiples of 2π to the argument as it will end up in the same place (2π = 360º)

2) De Moivre’s theorem [r (cosθ+isinθ)]n=r n (cosnθ+isinnθ )

WB18Solve the equation z3 = 1 and represent your solutions on an Argand diagram

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FMA2 Complex numbers WB19 Solve the equation z4 - 2√3i = 2 Give your answers in both the modulus-argument and

exponential forms.

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FMA2 Complex numbers Summary points

1. Make sure you get the statement of De Moivre’s theorem rightDe Moivre’s theorem says [(cosθ+isinθ)]n=cosnθ+i sin nθ that for all integers n. It does not say, for example, that cosnθ+ isinnθ=cosnθ+i sin nθ for all integers n. This is just one of numerous possible silly errors.2. Remember to deal with the modulus when using De Moivre’s theorem to find a power of a complex number

For example in [3(cos π3+isin π

3)]

5

=35(cos 5π3

+isin 5π3 ) a common mistake is to forget to

raise 3 to the power of 5.3. Make sure that you don’t get the modulus of an nth root of a complex number wrong Remember that |zn|=|z|n, and this applies not just to integer values of n, but includes rational values of n, as when taking roots of z.

4. Make sure that you get the right number of nth roots of a complex number There should be exactly nof them. Remember two complex numbers which have the same moduli and arguments which differ by a multiple of 2π are actually the same number

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