Name: cofunction identities and complementary angles
Uploaded: 2025-11-26T05:44:26.000Z
Duration: 1 min 27 s
Channel: JoVE

Les identités de cofonction décrivent comment les paires de fonctions trigonométriques sont liées lorsque la somme de leurs angles est de 90 degrés.Les deux identités de cofonction de base peuvent être prouvées en considérant un point sur le cercle trigono.Soit thêta un angle avec l’axe des x positif. Par définition, les coordonnées x et y sont alors respectivement le cosinus thêta et le sinus thêta.Pour le même point, mesurez son angle par rapport à l’axe des y positifs, 90 degrés moins thêta.Les coordonnées du point sont égales à sin 90 moins thêta et cos 90 moins thêta.Comme les coordonnées sont inchangées, l’équivalence des expressions prouve les identités de cofonction du sinus et du cosinus.Cependant, toutes les autres fonctions trigonométriques, qui sont basées sur le sinus et le cosinus, héritent de cette propriété.Ces identités aident aux calculs du monde réel.Par exemple, lorsqu’un bateau est loin d’un phare, la distance est mesurée à l’aide de l’angle d’élévation entre le bateau en mouvement et le sommet du phare. Les identités de cofonction permettent de remplacer la tangente de thêta par la tangente de l’angle complémentaire dans la formule de distance, en préservant le résultat avec une flexibilité dans les calculs.

identités trigonométriques iii

Trigonometric Identities III

Les identit&#233;s de cofonction sont un concept cl&#233; en trigonom&#233;trie. Elles d&#233;crivent la relation entre des fonctions trigonom&#233;triques lorsque leurs angles sont compl&#233;mentaires, c&#8217;est-&#224;-dire que leur somme vaut 90 &#176;. Sur le cercle trigonom&#233;trique, chaque angle &#952;, mesur&#233; dans le sens inverse des aiguilles d&#8217;une montre &#224; partir de l&#8217;axe des abscisses orient&#233; positivement, correspond &#224; un point de coordonn&#233;es (cos &#952;, sin &#952;). Ces valeurs repr&#233;sentent les composantes horizontale et verticale de la demi-droite terminale de l&#8217;angle.Si le m&#234;me point du cercle trigonom&#233;trique est d&#233;crit par l&#8217;angle compl&#233;mentaire 90 &#176; &#8722; &#952;, la r&#233;f&#233;rence se place alors sur l&#8217;axe des ordonn&#233;es positif. Dans ce cas, le segment horizontal initialement adjacent &#224; &#952; devient le c&#244;t&#233; oppos&#233; &#224; 90 &#176; &#8722; &#952;, tandis que le segment vertical, auparavant oppos&#233;, devient adjacent. Ainsi, les coordonn&#233;es s&#8217;&#233;crivent &#233;galement (sin (90 &#176; &#8722; &#952;), cos (90 &#176; &#8722; &#952;)). Puisque ces deux descriptions renvoient au m&#234;me point, il s&#8217;ensuit que&#160;:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1425/74;' src='https://cloudfront.jove.com/files/ftp_upload/41457/41457_Equation_1_editor_41457.svg' alt='41457_Equation_1.svg' /></figure>Ces &#233;galit&#233;s constituent les identit&#233;s fondamentales de cofonction, illustrant l&#8217;&#233;change des r&#244;les du sinus et du cosinus pour des angles compl&#233;mentaires.Au-del&#224; du sinus et du cosinus, le m&#234;me raisonnement s&#8217;&#233;tend aux autres fonctions trigonom&#233;triques, ce qui donne l&#8217;ensemble complet des identit&#233;s de cofonction :<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1444/361;' src='https://cloudfront.jove.com/files/ftp_upload/41457/41457_Equation_2_editor_41457.svg' alt='41457_Equation_2.svg' /></figure>Ces identit&#233;s mettent en &#233;vidence la sym&#233;trie des fonctions trigonom&#233;triques et leur lien &#233;troit avec les angles compl&#233;mentaires, tant dans les triangles rectangles que sur le cercle trigonom&#233;trique. Elles offrent une m&#233;thode syst&#233;matique pour passer d&#8217;une fonction &#224; une autre lorsque l&#8217;on travaille avec des angles mesur&#233;s par rapport &#224; des directions de r&#233;f&#233;rence distinctes.

Identités trigonométriques III

Les identit&#233;s de cofonction sont un concept cl&#233; en trigonom&#233;trie. Elles d&#233;crivent la relation entre des fonctions ...

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Math Fundamentals

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Trigonometry

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cofunction identities and complementary angles

Cofunction identities show how trigonometric functions change for complementary angles. Complementary angles add up to 90°. These identities are a useful part of trigonometry for grades 9–12.

The idea becomes clear on the unit circle. Every angle θ, measured counterclockwise from the positive x-axis, matches a point with coordinates (cos θ, sin θ). The cosine gives the horizontal component, and the sine gives the vertical component of the terminal side.

The same point can also be described using the complementary angle 90° − θ. In that form, the reference changes to the positive y-axis. The horizontal segment that was adjacent to θ becomes opposite to 90° − θ, and the vertical segment that was opposite becomes adjacent.

Because both descriptions refer to the same point, the coordinates can be written as (sin(90° − θ), cos(90° − θ)). This leads to the basic cofunction identities, where sine and cosine switch roles for complementary angles. The same pattern also applies to the other trigonometric functions, giving the full set of cofunction identities.

These identities show the symmetry of trigonometric functions. They also connect right-triangle relationships with the unit circle. That makes them a systematic way to change from one trig function to another when angles are measured from different reference directions.

Cofunction Identities and Complementary Angles

cofunction-identities-and-complementary-angles

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Cofunction identities show how trigonometric functions change for complementary angles. Complementary angles add up to 90°. These identities are a useful part of trigonometry for grades 9–12. The idea becomes clear on the unit circle. Every angle θ, measured counterclockwise from the positive x-axis, matches a point with coordinates (cos θ, sin θ). The cosine gives the horizontal component, and the sine gives the vertical component of the terminal side. The same point can also be described...

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key terms and concepts

Learn how cofunction identities connect sine, cosine, and other trig functions for complementary angles on the unit circle and in right triangles.

Cofunction Identities and Complementary Angles | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Cofunction Identities and Complementary Angles

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