Mynd:Academ Base of trigonometry.svg
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English: Elementary trigonometry is based on this assert: if two right triangles have equal acute angles, they are similar, so their side lengths are proportional. This proportionality is the origin of trigonometry, because of the proportionality constants between side lengths of such triangles. Three constants are written within the image, where θ is the common measure of five acute angles. For example, tan θ is a proportionality constant: the ratio of leg lengths of all right triangles similar to ABC, provided that the numerator of the ratio is the leg opposite to the angle that measures θ.
In a right triangle with an angle measuring θ, a ratio of two side lengths only depends on the position of the sides relative to the angle that measures θ. How to remember the position of each side relative to this angle, how to know assuredly the trigonometry formulas? Here is a traditional mnemonic, like a magic formula to provide access to the three equalities: SOH CAH TOA. Two similar right triangles are either directly or indirectly similar, except in case of isosceles triangles. Within the image, they are directly similar if they have the same colour, otherwise they are indirectly similar. To be clear, imagine a given triangle duplicated on a transparent tracing paper, to some scale. If the original triangle and its copy are isosceles, the duplicate is the same on the transparent sheet, whatever the side that we see: the front side or back side of the sheet, because of a mirror symmetry of the figure. Such isosceles triangles are said both directly and indirectly similar. Otherwise, the duplicate is either directly similar to the original or indirectly similar, depending on the side of the transparent sheet that we see. To be more strict, a given similarity is either direct or indirect, according to the sign of its determinant. For example, see the case of a determinant equal to –1.Français : La trigonométrie élémentaire est basée sur cette affirmation : si deux triangles rectangles ont des angles aigus égaux, alors ils sont semblables, de sorte que les longueurs de leurs côtés sont proportionnelles. Cette proportionnalité est à l’origine de la trigonométrie, à cause des coefficients de proportionnalité entre les longueurs des côtés de tels triangles. Trois coefficients sont écrits dans l’image, où θ est la mesure commune de cinq angles aigus. Par exemple, tan θ est un coefficient de proportionnalité : le rapport des côtés de l’angle droit de tous les triangles semblables à ABC, à condition que le numerateur du rapport soit le côté opposé à l’angle qui mesure θ.
Dans un triangle rectangle avec un angle de mesure θ, un rapport de deux longueurs de côtés dépend uniquement de la position des côtés par rapport à l’angle de mesure θ. Comment retenir la position de chaque côté relative à cet angle, comment être sûr des formules de trigonométrie ? Il existe un moyen mnémotechnique traditionnel, tel une formule magique donnant accès aux trois égalités : SOH CAH TOA (Sinus : Opposé sur Hypoténuse, Cosinus : Adjacent sur Hypoténuse, Tangente : Opposé sur Adjacent). Deux triangles rectangles semblables sont soit directement soit indirectement semblables, sauf dans le cas de triangles isocèles. Dans l’image, ils sont directement semblables s’ils ont la même couleur, sinon ils sont indirectement semblables. Pour être clair, imaginons un triangle donné reproduit sur un papier transparent, à une certaine échelle. Si le triangle original et sa copie sont isocèles, la reproduction est la même sur la feuille transparente, quel que soit le côté de la feuille que nous voyons : le recto ou le verso de la feuille, à cause d’une symétrie axiale de la figure. De tels triangles isocèles sont dits à la fois directement et indirectement semblables. Sinon, la reproduction est soit directement soit indirectement semblable, selon le côté de la feuille transparente que nous voyons. Pour être plus exact, une similitude est soit directe, soit indirecte, en fonction du signe de son déterminant. Par exemple, voir le déterminant d’une isométrie indirecte, égal à –1. |
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| Dagsetning/Tími | Smámynd | Víddir | Notandi | Athugasemd | |
|---|---|---|---|---|---|
| núverandi | 21. október 2012 kl. 09:00 | | 600 × 600 (3 KB) | Baelde | better presentation |
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