19.Simplify (i)( 1 +j
√
3 )^6 +( 1 −j√
3 )^6 and (ii) (√
3 −j)^15 by using De Moivre’s
theorem.20.Simplify
(cos 3θ−jsin 3θ)−^3 (cos 2θ+jsin 2θ)^4
(cos 7θ+jsin 7θ)^5 (cos 5θ−jsin 5θ)−^4
21.Determine all the roots in each case:
(i) Square roots of 1 (ii) Square roots of− 1
(iii) Square roots ofj (iv) Cube roots of−j
(v) Square roots of( 1 +j) (vi) Square roots of 1+√
3 j
(vii) Fourth roots of 1−j (viii) Fourth roots of√
3 +j22.Find the values ofzsatisfying
z^4 + 1 = 023.Simplify each of the following numbers to thea+jbform:
(i)cos 4α+jsin 4α
cos 3α−jsin 3α(ii)1
sin 2α+jcos 2α(iii)cos 2α−jsin 2α
sin 3α+jcos 3α24.Find the polar forms of the fourth roots of−16 and indicate the results on the Argand
diagram.
12.10 Applications
1.Ify=AejαtwhereAandαare constants, show that
d^2 y
dt^2+α^2 y=0. By assuming a
solution of the formy=eλt, find solutions for the differential equationd^2 y
dt^2+ 2dy
dt+ 2 y= 0by obtaining an equation forλ. This is the approach we will follow in Chapter 15.- (i) By equating the real parts of cos 5θ+jsin 5θ=(cosθ+jsinθ)^5 show that
cos 5θ=16 cos^5 θ−20 cos^3 θ+5cosθ. Also express sin 5θas powers of sinθ.
(ii) By using sinθ=ejθ−(ejθ)−^1
2 jand cosθ=ejθ+(ejθ)−^1
2express in terms ofsines and cosines of multiple angles (a) sin^4 θ,(b)cos^5 θ.Use(a)toevaluate
∫π/ 30sin^4 θdθ.(iii) Write cosθ=R(ejθ)and hence sum the series cosθ+cos 2θ+···+cosnθ.