Page:Annales de mathématiques pures et appliquées, 1812-1813, Tome 3.djvu/26

20

FORMULES

$2^{m}.P\left\{\operatorname {Sin} .^{2}{\tfrac {2(1\ldots m)-1}{2m}}{\tfrac {\varpi }{2}}\right\}=P\left\{\operatorname {Sin} .2{\tfrac {2(1\ldots m)-1}{2m}}{\tfrac {\varpi }{2}}\right\}\,;$

ou, en se rappelant que $\operatorname {Sin} .2x=2\operatorname {Sin} .x\operatorname {Cos} .x$

$2^{m}.P\left\{\operatorname {Sin} .^{2}{\tfrac {2(1\ldots m)-1}{2m}}{\tfrac {\varpi }{2}}\right\}=P\left\{2\operatorname {Sin} .{\tfrac {2(1\ldots m)-1}{2m}}{\tfrac {\varpi }{2}}\operatorname {Cos} .{\tfrac {2(1\ldots m)-1}{2m}}{\tfrac {\varpi }{2}}\right\}\,;$

ou, en remarquant que 2 est $m$ fois facteur dans le second membre,

$P\left\{\operatorname {Sin} .^{2}{\tfrac {2(1\ldots m)-1}{2m}}{\tfrac {\varpi }{2}}\right\}=P\left\{\operatorname {Sin} .{\tfrac {2(1\ldots m)-1}{2m}}{\tfrac {\varpi }{2}}\operatorname {Cos} .{\tfrac {2(1\ldots m)-1}{2m}}{\tfrac {\varpi }{2}}\right\}\,;$

ou encore

$P\left\{\operatorname {Sin} .{\tfrac {2(1\ldots m)-1}{2m}}{\tfrac {\varpi }{2}}\right\}P\left\{\operatorname {Sin} .{\tfrac {2(1\ldots m)-1}{2m}}{\tfrac {\varpi }{2}}\right\}=\ldots$

$P\left\{\operatorname {Sin} .{\tfrac {2(1\ldots m)-1}{2m}}{\tfrac {\varpi }{2}}\right\}P\left\{\operatorname {Cos} .{\tfrac {2(1\ldots m)-1}{2m}}{\tfrac {\varpi }{2}}\right\}\,;$

ou enfin

$P\left\{\operatorname {Sin} .{\tfrac {2(1\ldots m)-1}{2m}}{\tfrac {\varpi }{2}}\right\}=P\left\{\operatorname {Cos} .{\tfrac {2(1\ldots m)-1}{2m}}{\tfrac {\varpi }{2}}\right\}\,;\quad \mathrm {(C)}$

d’où résulte encore,

$P\left\{\operatorname {Tang} .{\tfrac {2(1\ldots m)-1}{2m}}{\tfrac {\varpi }{2}}\right\}=1.$

Posant ${\frac {\varpi }{4m}}=\omega ,$ d’où $m={\frac {\varpi }{4\omega }},$ et $2m-1={\frac {\varpi -2\omega }{2\omega }},$ il viendra, en substituant dans l’équation (C) et développant,

$\operatorname {Sin} .\omega \operatorname {Sin} .3\omega \operatorname {Sin} .5\omega \ldots \operatorname {Sin} .({\tfrac {1}{2}}\varpi -\omega )=\ldots$

$\operatorname {Cos} .\omega \operatorname {Cos} .3\omega \operatorname {Cos} .5\omega \ldots \operatorname {Cos} .({\tfrac {1}{2}}\varpi -\omega )\,;\quad \mathrm {(D)}$

équation qui, au surplus, se vérifie aisément d’elle-même, en observant que

${\begin{array}{lll}&\operatorname {Sin} .\omega &=\operatorname {Cos} .({\tfrac {1}{2}}\varpi -\omega ),\\&\operatorname {Sin} .3\omega &=\operatorname {Cos} .({\tfrac {1}{2}}\varpi -3\omega ),\\\end{array}}$

. . . . . . . . . . . . . . . . . . . . . . . . .

${\begin{array}{lll}&\operatorname {Sin} .({\tfrac {1}{2}}\varpi -3\omega )&=\operatorname {Cos} .3\omega ,\\&\operatorname {Sin} .({\tfrac {1}{2}}\varpi -\omega )&=\operatorname {Cos} .\omega .\\\end{array}}$