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FMA2 Complex numbers Objective Deadlines / Progress
Rev
iew
mod
ulus
arg
umen
t
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
Rec
ipro
cal a
nd
inve
rse
func
tions
Add
ition
form
ulas
1
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
2
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 ))
3
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)
4
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 )
5
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
6
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
7
FMA2 Complex numbers
8
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 ?
9
FMA2 Complex numbers WB10
Simplify the following: (cos 9 π
17+isin 9 π
17 )5
(cos 2π17
−isin 2π17 )
3
WB11 Express the following in the form x + iy where x Є R and y Є R(1+√3 i)7
10
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
11
FMA2 Complex numbers
WB13 Express the following as powers of cosθ: sin 6θsinθ
12
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
13
FMA2 Complex numbers
14
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
15
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
16
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
17
FMA2 Complex numbers WB19 Solve the equation z4 - 2√3i = 2 Give your answers in both the modulus-argument and
exponential forms.
18
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
19