If-z-1-1-then-prove-that-arg-z-1-2-arg-z-1-

Question Number 58592 by rahul 19 last updated on 25/Apr/19

I f ∣ z - 1 ∣= 1, t h e n p r o v e t h a t a r g (z) =

\frac{1}{2} a r g (z - 1) .

Answered by tanmay last updated on 26/Apr/19

z = {r e}^{i θ} = r c o s θ + i r s i n θ

z - 1 = (r c o s θ - 1) + i r s i n θ

∣ z - 1 ∣= \sqrt{{(r c o s θ - 1)}^{2} + {(r s i n θ)}^{2}}

r^{2} {c o s}^{2} θ - 2 r c o s θ + 1 + r^{2} {s i n}^{2} θ = 1

r^{2} - 2 r c o s θ = 0

r (r - 2 c o s θ) = 0

r \neq 0 r = 2 c o s θ

z = 2 c o s θ \times c o s θ + i 2 c o s θ s i n θ

A + i B = (2 {c o s}^{2} θ) + i (2 s i n θ c o s θ)

a r g (z) = θ

n o w

z - 1

2 {c o s}^{2} θ + i s i n 2 θ - 1

= c o s 2 θ + i s i n 2 θ

\frac{1}{2} a r g (z - 1)

\frac{1}{2} \times 2 θ = θ

Commented by JDamian last updated on 25/Apr/19

T h e r e i s a m i s t a k e i n t h e f o u r t h l i n e :

{r s i n}^{2} θ s h o u l d b e r^{2} {s i n}^{2} θ

Commented by tanmay last updated on 26/Apr/19

t h a n k y o u s i r

Commented by rahul 19 last updated on 26/Apr/19

t h a n k U s i r .