082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

350 Chapter 9 Complex numbers
Example9.22 Show thatcos 2 θ = 1 (cos2θ +1).
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Solution The formulae obtained in Example 9.21 allow us to obtain expressions for powers of
cosθ andsinθ.Since2cosθ =z+ 1 , squaring both sides wehave
z
(
2 2 cos 2 θ = z + 1 ) 2
=z 2 +2+ 1 z z 2
=
(z 2 + 1 )
+2
z 2
Butz 2 + 1 z =2cos2θ,so 2
2 2 cos 2 θ =2cos2θ+2
and therefore,
cos 2 θ = 1 2 cos2θ + 1 2
as required.
= 1 (cos2θ +1)
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EXERCISES9.9
1 Express (cos θ +jsinθ) 9 and (cos θ +jsinθ) 1/2 in
the formcosnθ +jsinnθ.
2 UseDeMoivre’stheoremto simplify
(a) (cos3θ +jsin3θ)(cos4θ +jsin4θ)
cos8θ +jsin8θ
(b)
cos2θ −jsin2θ
3 Solve the equations
(a) z 3 +1=0
(b) z 4 =1+j
4 Find 3 √ 2 +2j and displayyoursolutionsonan
Argand diagram.
5 Prove the following trigonometricidentities:
(a) cos4θ =8cos 4 θ−8cos 2 θ+1
(b) 32sin 6 θ = 10 −15cos2θ +6cos4θ −cos6θ
6 Solvethe equationz 4 +25 = 0.
7 Findthe fifthrootsofjanddepictyoursolutionson
an Arganddiagram.
8 Show thatcos 3 θ = 1 (cos3θ +3cosθ).
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9 Show thatsin 4 θ = 1 (cos4θ −4cos2θ +3).
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10 Expresscos5θ in termsofpowers ofcos θ.
11 Expresssin5θ in terms ofpowersofsin θ.
12 Given e jθ = cos θ +jsin θ,proveDeMoivre’s
theoremin the form
(e jθ ) n =cosnθ+jsinnθ
Solutions
1 cos9θ +jsin 9θ,cos(θ/2) +jsin(θ/2)
2 (a) cos 7θ +jsin 7θ
(b) cos 10θ +j sin 10θ
3 (a)1̸ π/3 +2nπ/3 n = 0,1,2
(b) 2 1/8̸ π/16 +nπ/2 n = 0,1,2,3
4 8 1/6̸ π/12 +2nπ/3 n = 0,1,2