Question

If z1 & z2 are two non-zero complex numbers, then prove that arg z1z2 = arg z1 - arg z2.

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Answer 1

Answer:

Let

$z_{1} = a_{1} {e}^{i \theta_1}$

$z_{2} = a_{2} {e}^{i \theta_2}$

$z_{1} z_{2} = a_{1} {e}^{i \theta_1} a_{2} {e}^{i \theta_2} = a_{1} a_{2} {e}^{i (\theta_1 + \theta_2)}$

Hence proved

Answer 2

Answer:

Let z

1

=cosθ

1

+isinθ

1

z

2

=cosθ

2

+isinθ

2

∴z

1

+z

2

=(cosθ

1

+cosθ

2

)+i(sinθ

1

+sinθ

2

)

Now,

∣z

1

+z+2∣=∣z

1

∣+∣z

2

∣

⇒

(cosθ

1

)

2

+(sinθ

1

+sinθ

2

)

2

=1+1

On squaring both side

⇒2(1+cos(θ

1

−θ

2

))=4

⇒cos(θ

1

−θ

2

)=1

⇒θ

1

−θ

2

=0(∵cos0=1)

⇒argz

1

−argz

2

=0