Page:Annales de mathématiques pures et appliquées, 1824-1825, Tome 15.djvu/67

2.^o Si $x$ est compris entre $4n\varpi +{\frac {3}{2}}\varpi$ et $4n\varpi +{\frac {5}{2}}\varpi$ on aura

$\operatorname {Sin} .{\frac {1}{2}}x+\operatorname {Cos} .{\frac {1}{2}}x=-{\sqrt {1+\operatorname {Sin} .x}},\quad \operatorname {Sin} .{\frac {1}{2}}x-\operatorname {Cos} .{\frac {1}{2}}x=+{\sqrt {1-\operatorname {Sin} .x}}\,;$

et par suite

${\begin{aligned}\operatorname {Sin} .{\frac {1}{2}}x&={\frac {1}{2}}\left(-{\sqrt {1+\operatorname {Sin} .x}}+{\sqrt {1-\operatorname {Sin} .x}}\right),\\\\\operatorname {Cos} .{\frac {1}{2}}x&={\frac {1}{2}}\left(-{\sqrt {1+\operatorname {Sin} .x}}-{\sqrt {1-\operatorname {Sin} .x}}\right).\end{aligned}}$

3.^o Si $x$ est compris entre $4n\varpi +{\frac {5}{2}}\varpi$ et $4n\varpi +{\frac {7}{2}}\varpi$ on aura

$\operatorname {Sin} .{\frac {1}{2}}x+\operatorname {Cos} .{\frac {1}{2}}x=-{\sqrt {1+\operatorname {Sin} .x}},\quad \operatorname {Sin} .{\frac {1}{2}}x-\operatorname {Cos} .{\frac {1}{2}}x=-{\sqrt {1-\operatorname {Sin} .x}}\,;$

et par suite

${\begin{aligned}\operatorname {Sin} .{\frac {1}{2}}x&={\frac {1}{2}}\left(-{\sqrt {1+\operatorname {Sin} .x}}-{\sqrt {1-\operatorname {Sin} .x}}\right),\\\\\operatorname {Cos} .{\frac {1}{2}}x&={\frac {1}{2}}\left(-{\sqrt {1+\operatorname {Sin} .x}}+{\sqrt {1-\operatorname {Sin} .x}}\right).\end{aligned}}$

4.^o Si $x$ est compris entre $4n\varpi +{\frac {7}{2}}\varpi$ et $4n\varpi +{\frac {9}{2}}\varpi$ on aura

$\operatorname {Sin} .{\frac {1}{2}}x+\operatorname {Cos} .{\frac {1}{2}}x=+{\sqrt {1+\operatorname {Sin} .x}},\quad \operatorname {Sin} .{\frac {1}{2}}x-\operatorname {Cos} .{\frac {1}{2}}x=-{\sqrt {1-\operatorname {Sin} .x}}\,;$

et par suite

${\begin{aligned}\operatorname {Sin} .{\frac {1}{2}}x&={\frac {1}{2}}\left(+{\sqrt {1+\operatorname {Sin} .x}}-{\sqrt {1-\operatorname {Sin} .x}}\right),\\\\\operatorname {Cos} .{\frac {1}{2}}x&={\frac {1}{2}}\left(+{\sqrt {1+\operatorname {Sin} .x}}+{\sqrt {1-\operatorname {Sin} .x}}\right).\end{aligned}}$

M. Legendre donne uniquement ces dernières formules, et les