Question

Show that the folowing complex numbers have the same argument.

a. z1 = 3 + 3√3i and z2 = 1 + √3i
b. z1 = 1 + i and z2 = 4 + 4i

Answer 1

Answer and explanation

(a) We have given that
`z_1=3+3√(3)i`

And
`z_2=1+√(3)i`

We have to prove that both complex number has same argument

Argument is given by
`\Theta =tan^(-1)((imaginary\ part)/(real\ part))`

Argument of
`z_1=tan^(-1)(3√(3))/(3)=tan^(-1)√(3)=60^(\circ)`

Argument of
`z_2=tan^(-1)(√(3))/(1)=tan^(-1)√(3)=60^(\circ)`

Hence both
`z_!\ and\ z_2` have same argument

(b) We have given We have given that
`z_1=1+i`

And
`z_2=4+4i`

We have to prove that both complex number has same argument

Argument is given by
`\Theta =tan^(-1)((imaginary\ part)/(real\ part))`

Argument of
`z_1=tan^(-1)(1)/(1)=tan^(-1)1=45^(\circ)`

Argument of
`z_2=tan^(-1)(4)/(4)=tan^(-1)1=45^(\circ)`

Hence both
`z_!\ and\ z_2` have same argument