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Question Number 81954 by Cmr 237 last updated on 16/Feb/20

s o i t α \in] 0; π [. d e t e r m i n e r :

1) l e m o d u l e e t l^{'} a r g u m e n t d e :

a) 1 - e^{i α}, b) 1 + e^{i α}

2) d e d u i r e l e m o d u l e e t l^{'} a r g u m e n t d e

a) \frac{1 - e^{i α}}{1 + e^{i α}}, b) (1 - e^{i α}) (1 + e^{i α})

r o c h i n e l 930 @ g m a i l . c

Commented by abdomathmax last updated on 16/Feb/20

z = 1 - e^{i α} = 1 - c o s α - i s i n α = 2 {s i n}^{2} (\frac{α}{2}) - 2 i s i n (\frac{α}{2}) c o s (\frac{α}{2})

= 2 s i n (\frac{α}{2}) (s i n (\frac{α}{2}) - i c o s (\frac{α}{2}))

0 < \frac{α}{2} < \frac{π}{2} \Rightarrow s i n (\frac{α}{2}) > 0 \Rightarrow z = 2 s i n (\frac{α}{2}) {c o s (\frac{π}{2} - \frac{α}{2}) - i s i n (\frac{π}{2} - \frac{α}{2})}

= 2 s i n (\frac{α}{2}) {c o s (\frac{α - π}{2}) + i s i n (\frac{α - π}{2})} \Rightarrow

∣ z ∣= 2 s i n (\frac{α}{2}) a n d a r g (z) = \frac{α - π}{2} [2 π]

Commented by abdomathmax last updated on 16/Feb/20

Z = 1 + e^{i α} \Rightarrow Z = 1 + c o s α + i s i n α

= 2 {c o s}^{2} (\frac{α}{2}) + 2 i s i n (\frac{α}{2}) c o s (\frac{α}{2})

= 2 c o s (\frac{α}{2}) {c o s (\frac{α}{2}) + i s i n (\frac{α}{2})}

c o s (\frac{α}{2}) > 0 d u e t o 0 < α < π \Rightarrow∣ Z ∣= 2 c o s (\frac{α}{2}) a n d

a r g Z = \frac{α}{2} [2 π]

Commented by abdomathmax last updated on 16/Feb/20

∣ \frac{1 - e^{i α}}{1 + e^{i α}} ∣ = \frac{∣ 1 - e^{i α} ∣}{∣ 1 + e^{i α} ∣} = \frac{2 s i n (\frac{α}{2})}{2 c o s (\frac{α}{2})} = t a n (\frac{α}{2}) a n d

a r g (\frac{1 - e^{i α}}{1 + e^{i α}}) = a r g (1 - e^{i α}) - a r g (1 + e^{i α}) [2 π]

= \frac{α - π}{2} - \frac{α}{2} [2 π] = - \frac{π}{2} [2 π]

Commented by abdomathmax last updated on 16/Feb/20

∣ (1 - e^{i α}) (1 + e^{i α}) ∣= 2 s i n (\frac{α}{2}) \times 2 c o s (\frac{α}{2})

= 2 s i n (α) a n d a r g {(1 - e^{i α}) (1 + e^{i α})}

= a r g (1 - e^{i α}) + a r g (1 + e^{i α}) [2 π]

= \frac{α - π}{2} + \frac{α}{2} [2 π] = α - \frac{π}{2}

Commented by Cmr 237 last updated on 16/Feb/20

1) 1 + e^{i α} = e^{\frac{i α}{2}} (e^{- \frac{i α}{2}} + e^{\frac{i α}{2}})

= 2 c o s (\frac{α}{2}) e^{\frac{i α}{2}}

∣ 1 + e^{i α} ∣=∣ 2 c o s (\frac{α}{2}) e^{\frac{i α}{2}} ∣

= 2 c o s (\frac{α}{2})

∣ 1 - e^{i α} ∣=∣ - 2 i s i n (\frac{α}{2}) e^{\frac{i α}{2}} ∣

= 2 s i n (\frac{α}{2})

a r g (1 - e^{i α}) = a r g (- 2 i s i n (\frac{α}{2}) e^{\frac{i α}{2}})

= a r g (- 2 s i n (\frac{α}{2})) + a r g (i) + a r g (e^{\frac{i α}{2}})

= π + \frac{π}{2} + \frac{α}{2} + 2 k π

= \frac{3 π + α}{2} + 2 k π

Commented by mathmax by abdo last updated on 16/Feb/20

\frac{3 π + α}{2} + 2 k π = 2 π - \frac{π}{2} + \frac{α}{2} + 2 k π = \frac{α - π}{2} [2 π]