Understanding Engineering Mathematics

12.If|z 1 |=5, Argz 1 =π/3,|z 2 |=3, Argz 2 =π/4, find the Cartesian forms ofz 1 and
z 2 and the values of:

(i) |z 1 z 2 | (ii)

∣

∣

∣

∣

z 1

z 2

∣

∣

∣

∣ (iii) |z

2

1 |

(iv) Arg(z 1 z 2 ) (v) Arg

(

z 1

z 2

)

(vi) Argz ̄ 1

13.Show that multiplication byjrotates a complex number through

π

2

in the anticlock-

wise direction and division byjrotates it

π

2

in the clockwise direction.

14.Ifz 1 = 3

(π

6

)

, z 2 = 2

(π

18

)

, z 3 =^

(π

3

)

find the polar form of:

(i) z 1 z 2 (ii)

z 1

z 2

(iii) z 1 z 2 z 3

(iv)

z 2

z^23

(v)

z 2 z 3

z 1

(vi)

z^21 z 3

z 2

15.Evaluate the powers indicated by use of the polar form.

(i) ( 1 −j)^8 (ii) (

√

3 +j)^6 (iii) ( 2 + 2 j)^4

16.Plot the complex numbers

(i) 1−j (ii) 2j (iii)

√

3

2

+

1

2

j (iv) − 3 j

on the Argand diagram and put them in the formejθ.

17.Express the following numbers in the forma+jb:

(i) ejπ/^3 (ii) e−jπ/^6 (iii) e−(^1 +jπ)/^3 (iv)

ejπ/^3

j

(v)

e−jπ/^4

1 + 2 j

(vi)

e−jπ/^6

2 −j

18.Use the power series forexwithx=jθto findejθin the formA+jBwhereAand
Bare real power series inθ. Hence show that:

ejθ=cosθ+jsinθ

Use this result to prove that

(cosθ+jsinθ)n=cosnθ+jsinnθ

Hence evaluate

(

1

2

+j

√

3

2

) 43