Name: double-angle and half-angle formulas
Uploaded: 2025-11-26T05:44:27.000Z
Duration: 1 min 28 s
Channel: JoVE

Dans un skate park, un rider s’approche d’une rampe incurvée. Sa pente changeante modifie l’accélération du cycliste, traçant un chemin façonné par un mouvement angulaire.Au fur et à mesure que le cycliste monte ou descend la rampe, la pente du chemin change, avec des similitudes avec la fonction tangente, qui compare la montée verticale à la course horizontale.La tangente est définie comme le sinus divisé par le cosinus pour tout angle mesuré à partir de l’horizontale.À partir de cette définition, l’identité du double angle peut être dérivée. Rappelons le sinus et le cosinus de deux fois l’angle. En les substituant dans la définition de tangente, on obtient le résultat : tangente de deux fois l’angle. Lorsque le cycliste s’élance plus haut, et que l’angle double, cette identité prédit la nouvelle pente.Pour les petits angles près de la base de la rampe, où le cycliste entre dans une courbe douce, l’identité du demi-angle devient utile.En rappelant et en appliquant les identités sinus et demi-angle cosinus à la définition de la tangente, on obtient une expression. En le simplifiant en prenant son conjugué et en le multipliant par un cosinus plus ou un moins, on obtient deux formes équivalentes de l’identité tangente demi-angle.Ces identités aident à analyser les changements de direction et à modéliser le mouvement courbe dans la dynamique du skate park.

identités trigonométriques ii

Trigonometric Identities II

Les identit&#233;s trigonom&#233;triques d&#8217;angle double et de demi-angle d&#233;coulent des formules fondamentales de somme et de diff&#233;rence et constituent des outils essentiels pour simplifier des expressions, r&#233;soudre des &#233;quations et &#233;valuer des int&#233;grales. Elles r&#233;duisent la complexit&#233; des fonctions trigonom&#233;triques en reliant des fonctions d&#8217;un angle multiple ou fractionnaire &#224; des fonctions d&#8217;un angle simple. Leurs applications s&#8217;&#233;tendent aux math&#233;matiques, &#224; la physique et &#224; l&#8217;ing&#233;nierie, notamment en analyse de Fourier, en m&#233;canique ondulatoire et en calcul diff&#233;rentiel et int&#233;gral.Les identit&#233;s d&#8217;angle double s&#8217;obtiennent en substituant des angles &#233;gaux dans les formules de somme. Pour le sinus et le cosinus, elles sont&#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:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/41035/41035_Equation_1_editor_41035.svg' alt='41035_Equation_1.svg' /></figure>Pour la tangente, l&#8217;identit&#233; d&#8217;angle double est&#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:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/41035/41035_Equation_2_editor_41035.svg' alt='41035_Equation_2.svg' /></figure>Ces identit&#233;s sont particuli&#232;rement utiles pour abaisser le degr&#233; (r&#233;duire les puissances &#233;lev&#233;es) des fonctions trigonom&#233;triques et pour simplifier des int&#233;grales.Les identit&#233;s de demi-angle se d&#233;duisent des formules d&#8217;angles doubles en isolant sin^2&#952; et cos^2&#952;. Elles s&#8217;expriment ainsi&#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:1920/650;' src='https://cloudfront.jove.com/files/ftp_upload/41035/41035_Equation_3_editor_41035.svg' alt='41035_Equation_3.svg' /></figure>Le choix du signe d&#233;pend du quadrant dans lequel se situe le demi-angle. Ces identit&#233;s sont cruciales pour l&#8217;&#233;valuation des fonctions trigonom&#233;triques pour des angles non usuels et dans les techniques d&#8217;int&#233;gration, telles que la substitution trigonom&#233;trique.

Identités trigonométriques II

Les identit&#233;s trigonom&#233;triques d&#8217;angle double et de demi-angle d&#233;coulent des formules fondamentales de somme et de diff&#233;rence et ...

Education

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JoVE Core

Math Fundamentals

Fondamentaux des mathématiques

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Trigonometry

Trigonométrie

KEY TERMS AND CONCEPTS

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double-angle and half-angle formulas

Double-angle and half-angle trigonometric formulas connect angles that are multiplied or divided by two. They come from the sum and difference formulas, so they build on earlier trigonometry skills. These identities help simplify expressions, solve equations, and evaluate integrals.

The double-angle formulas come from substituting equal angles into the sum formulas. For sine and cosine, they link the value at 2θ to the value at θ. The tangent double-angle identity follows the same idea. These formulas are useful when simplifying higher powers of trigonometric expressions.

The half-angle formulas are found by solving the double-angle equations for sin 2θ and cos 2θ. They give trigonometric values for half of an angle instead of the whole angle. The sign in a half-angle formula depends on the quadrant where the half-angle lies.

These identities are important for finding trigonometric values for nonstandard angles. They also appear in calculus, especially in trigonometric substitution and in methods for evaluating integrals. Their uses also extend into mathematics, physics, and engineering, including Fourier analysis and wave mechanics.

Double-Angle and Half-Angle Formulas

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trigonometric identities ii

Double-angle and half-angle trigonometric formulas connect angles that are multiplied or divided by two. They come from the sum and difference formulas, so they build on earlier trigonometry skills. These identities help simplify expressions, solve equations, and evaluate integrals. The double-angle formulas come from substituting equal angles into the sum formulas. For sine and cosine, they link the value at 2θ to the value at θ. The tangent double-angle identity follows the same idea. These...

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key terms and concepts

Learn how double-angle and half-angle trig formulas come from sum and difference identities, and how they help simplify equations, powers, and integrals.

Double-Angle and Half-Angle Formulas | 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.

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Double-Angle and Half-Angle Formulas

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