Name: trig derivatives in ferris wheel motion
Uploaded: 2026-01-12T12:24:27.000Z
Duration: 1 min 26 s
Channel: JoVE

Imaginez une grande roue tournant à vitesse constante. La hauteur verticale d’un point sur son bord varie en douceur au fil du temps, suivant une onde sinusoïdale.Le taux de changement de hauteur du cavalier varie également de manière fluide et cyclique, comme une autre vague. Ce taux variable correspond à la dérivée de la fonction sinusoïdale.Pour trouver la dérivée du sinus x, on commence par la définition limite d’une dérivée et on l’applique au sinus x.D’abord, dévoloppons le sinus de (x + h) en utilisant la formule d’addition. Ensuite, réarrangez les termes et éliminez les sinus x et cosinus x.Lorsque h tend vers zéro, le cosinus h moins 1 sur h tend vers zéro, et le sinus h sur h tend vers 1.En substituant ces limites, on découvre que la dérivée du sinus x est le cosinus x.La même approche montre que la dérivée du cosinus x est un sinus négatif x.Puisque la fonction tangente est le rapport entre le sinus et le cosinus, pour trouver la dérivée, on applique la règle du quotient. Maintenant, en substituant les valeurs des dérivées des fonctions sinus et cosinus et en simplifiant à l’aide d’identités trigonométriques, on obtient la dérivée de la tangente comme une sécante au carré.

dérivées des fonctions trigonométriques

Derivatives of the Trigonometric Functions

Le mouvement d&#x2019;une grande roue tournant &#xE0; vitesse constante offre un mod&#xE8;le intuitif pour comprendre les fonctions trigonom&#xE9;triques et leurs d&#xE9;riv&#xE9;es. Lorsqu&#x2019;un passager se d&#xE9;place le long de la trajectoire circulaire, sa hauteur verticale au-dessus du sol varie de mani&#xE8;re continue et p&#xE9;riodique. Ce mouvement vertical peut &#xEA;tre repr&#xE9;sent&#xE9; avec pr&#xE9;cision par une fonction sinus, refl&#xE9;tant la r&#xE9;p&#xE9;tition des mont&#xE9;es et des descentes inh&#xE9;rente au mouvement circulaire.Hauteur et taux de variationSi la hauteur du passager est mod&#xE9;lis&#xE9;e par une fonction sinus, le taux de variation de cette hauteur correspond &#xE0; la d&#xE9;riv&#xE9;e de cette fonction. Ce taux de variation n&#x2019;est pas constant ; il varie de mani&#xE8;re continue et cyclique, atteignant des valeurs maximales lorsque le passager se d&#xE9;place le plus rapidement vers le haut ou vers le bas, et devient nul aux points les plus hauts et les plus bas de la trajectoire. Math&#xE9;matiquement, la d&#xE9;riv&#xE9;e de la fonction sinus est la fonction cosinus,\begin{equation*} \jfrac{d}{dx}(\sin x) = \cos x \end{equation*}qui rend compte de cette variation du taux de d&#xE9;placement vertical.D&#xE9;riv&#xE9;es trigonom&#xE9;triques associ&#xE9;esLa fonction cosinus d&#xE9;crit &#xE9;galement un comportement p&#xE9;riodique, mais avec un d&#xE9;phasage par rapport &#xE0; la fonction sinus. Sa d&#xE9;riv&#xE9;e refl&#xE8;te l&#x2019;&#xE9;volution de ses valeurs au cours du temps et est donn&#xE9;e par\begin{equation*} \jfrac{d}{dx}(\cos x) = -\sin x \end{equation*}Le signe n&#xE9;gatif indique que la fonction cosinus d&#xE9;cro&#xEE;t l&#xE0; o&#xF9; la fonction sinus cro&#xEE;t, et inversement, soulignant ainsi la relation &#xE9;troite entre ces deux fonctions.La fonction tangente, d&#xE9;finie comme le rapport du sinus au cosinus, mod&#xE9;lise des grandeurs telles que la pente dans des contextes p&#xE9;riodiques. Sa d&#xE9;riv&#xE9;e d&#xE9;coule directement de r&#xE9;sultats trigonom&#xE9;triques connus et s&#x2019;exprime par \begin{equation*} \jfrac{d}{dx}(\tan x) = \sec^2 x \end{equation*}Ensemble, ces relations de d&#xE9;riv&#xE9;es d&#xE9;montrent comment les fonctions trigonom&#xE9;triques d&#xE9;crivent &#xE0; la fois la position et le taux de variation dans des syst&#xE8;mes pr&#xE9;sentant un mouvement p&#xE9;riodique et continu, comme une grande roue tournant &#xE0; vitesse constante.

Dérivées des fonctions trigonométriques

Le mouvement d&#x2019;une grande roue tournant &#xE0; vitesse constante offre un mod&#xE8;le intuitif pour comprendre les fonctions trigonom&#xE9;triques ...

derivatives-of-the-trigonometric-functions

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trig derivatives in ferris wheel motion

Trig derivatives can be understood through the motion of a Ferris wheel. As the wheel turns at a constant speed, a rider’s height above the ground changes in a smooth, repeating pattern over time. That vertical motion can be modeled with a sine function, because it matches the rise and fall of circular motion.

The derivative of that height tells how fast the rider’s position is changing at each moment. This rate of change is not constant. It is greatest when the rider is moving fastest upward or downward. It becomes zero at the highest and lowest points of the ride.

For a sine function, the derivative is the cosine function. Cosine describes the changing rate of vertical motion as the rider moves around the wheel. The cosine graph is phase-shifted from sine, so the two functions line up differently over time.

The derivative of cosine is negative sine. The negative sign shows that cosine decreases where sine increases, and the reverse is also true. The tangent function, which is sine divided by cosine, is also used to model slope in periodic settings, and its derivative follows from known trigonometric results. Together, these relationships show how trigonometric functions describe both position and rate of change in smooth periodic motion.

Trig Derivatives in Ferris Wheel Motion

derivatives of simple functions

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the product rule

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the quotient rule

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derivatives of the trigonometric functions

Trig derivatives can be understood through the motion of a Ferris wheel. As the wheel turns at a constant speed, a rider’s height above the ground changes in a smooth, repeating pattern over time. That vertical motion can be modeled with a sine function, because it matches the rise and fall of circular motion. The derivative of that height tells how fast the rider’s position is changing at each moment. This rate of change is not constant. It is greatest when the rider is moving fastest upward...

limits on trigonometric functions

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the chain rule

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the number e as a limit

e from Natural Log Derivatives

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derivatives-of-inverse-trigonometric-functions

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application of rates of change

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hyperbolic functions

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inverse-hyperbolic-functions-and-their-derivatives

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key terms and concepts

Learn how sine, cosine, and tangent derivatives describe a Ferris wheel rider’s changing height and rate of motion in a clear high school science context.

Trig Derivatives in Ferris Wheel Motion | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Trig Derivatives in Ferris Wheel Motion

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