Question

Find the modulus and amplitude for each of the following complex numbers:
i) 7 - 5i
ii) $\sqrt{3}+\sqrt{2}i$
iii) -8 + 15i
iv) -3(1 - i)

Answer 1

Answer:

Step-by-step explanation:

Hi,

If z = a + ib is any complex number such that a ∈R, b∈R  

and i = √-1,

then the modulus of complex number, |Z| is given by

|Z| = √a² + b²

Amplitude of the complex number , θ is given by,

θ = tan⁻¹(b/a)

i) Z = 7 - 5i

Modulus,|Z| = √7² + 5² = √74

Amplitude,θ = tan⁻¹(-5/7) = -tan⁻¹(5/7)

ii) Z = √3 + i√2i

Modulus,|Z| = √(√3)² + (√5)² = √3 + 5 = √8

Amplitude,θ = tan⁻¹(√2/√3)

iii)  Z = -8 + 15i

Modulus,|Z| = √8² + 15² = √289 = 17

Amplitude,θ = tan⁻¹(15/-8) = -tan⁻¹(15/8)

iv) Z = -3(1 - i)

= -3 + 3i

Modulus,|Z| = √3² + 3² = √18 = 3√2

Amplitude,θ = tan⁻¹(3/-3) = -tan⁻¹(1) = -π/4

Hope, it helps !

Answer 2

Answer:

Modulus

(1) 7-5i

a=7

b=5

|Z|= √a^2+b^2

= √(7)^2+(5)^2

= √49+25

= √74

(2) √3+√2i

a=√3

b=√2

|Z|=√a^2+b^2

=√(√3)^2+(√2)^2

=√3+2

=√5

(3) -8+15i

a=(-8)

b=15

|Z|=√a^2+b^2

= (-8)^2+(15)^2

=√64+225

=√289

=17

(4) -3(1-i)

-3+I

a=-3

b=1

|Z|=√a^2+b^2

=√(-3)^2+(1)^2

=√9+1

=√10

Hope this will help you

Step-by-step explanation: