20
FORMULES
![{\displaystyle 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\}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3fd87fda65fd965180ce202e4d3b19efa896c86)
ou, en se rappelant que ![{\displaystyle \operatorname {Sin} .2x=2\operatorname {Sin} .x\operatorname {Cos} .x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69ff73657ab8ac3540f748f248c935e4cbafb259)
![{\displaystyle 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\}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c45e30025560617c7b8cbd6caec72d14633a0c23)
ou, en remarquant que 2 est
fois facteur dans le second membre,
![{\displaystyle 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\}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/507c3c850c0d315a71549ebc55d5aa03da2f174f)
ou encore
![{\displaystyle 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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ceed480869c00a94a45883a617bf0b40aa3c68c)
![{\displaystyle 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\}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cb49330d2730daa2e85dc4044f1c7be3a78b6c7)
ou enfin
![{\displaystyle 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)} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ef19ddd9d69ec13e1f4100d522817cfa0b1482)
d’où résulte encore,
![{\displaystyle P\left\{\operatorname {Tang} .{\tfrac {2(1\ldots m)-1}{2m}}{\tfrac {\varpi }{2}}\right\}=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f228fa01b4de865f33da4c7d20e5b72dcd387fa)
Posant
d’où
et
il viendra, en substituant dans l’équation (C) et développant,
![{\displaystyle \operatorname {Sin} .\omega \operatorname {Sin} .3\omega \operatorname {Sin} .5\omega \ldots \operatorname {Sin} .({\tfrac {1}{2}}\varpi -\omega )=\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/650e59b3efca1e40dd16d7215c0f062c3756d88e)
![{\displaystyle \operatorname {Cos} .\omega \operatorname {Cos} .3\omega \operatorname {Cos} .5\omega \ldots \operatorname {Cos} .({\tfrac {1}{2}}\varpi -\omega )\,;\quad \mathrm {(D)} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/726e69a98c0a0d56fb92cc54ee38a684e83c5337)
équation qui, au surplus, se vérifie aisément d’elle-même, en observant que
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2667f0754d9905b9b89742942a67938b3f9cd4a)
. . . . . . . . . . . . . . . . . . . . . . . . .
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53592c4031409df2956c1ae108155136e64f9e87)