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Linenn - Jeux et Mathematiques - <br /> <b>Deprecated</b>: Non-static method plxUtils::strCheck() should not be called statically in <b>/homepages/3/d170528928/htdocs/jm/bn/core/lib/class.plx.show.php</b> on line <b>127</b><br /> Angles entiers - linenn


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Angles entiers

Par
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jp, le
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mardi 08 juin 2010 à
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Géométrie | Page principale Jeux et Mathématiques

Certains triangles à côtés entiers $x$, $y$, $z$ permettent de construire  avec une bonne approximation des angles de valeurs données. Les données ci-dessous correspondent aux mesures entières en degrés de l'angle $a=$  1°, 2°, 3° etc. On pourrait très bien choisir d'autres valeurs.

L'angle $a$ a pour côtés $x$ et $y$ , tandis que le côté $z$ est opposé.

$\displaystyle \cos a = \frac{x^2 + y^2 - z^2}{2xy}$ et $z = \sqrt{x^2 + y^2 -2xy \cos a$


L'approximation est inférieure à la valeur indiquée dans la dernière colonne du tableau. Le côté $x$ est supérieur ou égal à $y$ et dans les tableaux on s'est limité à calculer les valeurs $x \le 1000$.
On obtient une très bonne approximation par défaut d'un angle de $1$ degré en prenant $x=5\,101$, $y=2\,871$, $z=2\,231$, l'erreur est inférieure à $0.000\,000\,004$ degré.

La figure suivante montre le partage d'un angle de 360° utilisant deux cercles concentriques de rayons $x$ et $y$ à l'aide d'un compas dont l'écartement est $z$. Pour un angle $a=24$degrés, on peut prendre par exemple les rayons $x=75$mm, $y=63$mm et l'ouverture de compas $z=31$mm, ces valeurs se trouvent dans l'un des tableaux.

 

 

 

 

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