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

En un parque de patinaje, un ciclista se acerca a una rampa curva. Su pendiente cambiante altera la aceleración del ciclista, trazando un camino moldeado por el movimiento angular.A medida que el ciclista sube o baja por la rampa, la inclinación del camino cambia, con similitudes con la función tangente, que compara la subida vertical con la carrera horizontal.La tangente se define como seno dividido por coseno para cualquier ángulo medido desde la horizontal.A partir de esta definición, se puede derivar la identidad de doble ángulo. Recuerde el seno y el coseno del doble del ángulo. Sustituirlos en la definición de tangente conduce al resultado: tangente del doble del ángulo. Cuando el ciclista se lanza más alto y el ángulo se duplica, esta identidad predice la nueva pendiente.Para ángulos más pequeños cerca de la base de la rampa, donde el ciclista entra en una curva suave, la identidad de medio ángulo se vuelve útil.Recordar y aplicar las identidades de semiángulo seno y coseno a la definición tangente da una expresión. Simplificándolo tomando su conjugado y multiplicando por uno más o menos coseno produce dos formas equivalentes de la identidad tangente de semiángulo.Estas identidades ayudan a analizar los cambios direccionales y modelar el movimiento curvo en la dinámica del parque de patinaje.

identidades trigonométricas ii

Trigonometric Identities II

Las identidades trigonom&#233;tricas de &#225;ngulo doble y de medio &#225;ngulo se derivan de las f&#243;rmulas fundamentales de suma y diferencia y constituyen herramientas esenciales para simplificar expresiones, resolver ecuaciones y evaluar integrales. Estas identidades reducen la complejidad de las funciones trigonom&#233;tricas al relacionar funciones de un &#225;ngulo m&#250;ltiplo o fraccionario con funciones de un &#225;ngulo simple. Sus aplicaciones se extienden a las matem&#225;ticas, la f&#237;sica y la ingenier&#237;a, particularmente al an&#225;lisis de Fourier, la mec&#225;nica ondulatoria y el c&#225;lculo.Las identidades de &#225;ngulo doble se obtienen sustituyendo &#225;ngulos iguales en las f&#243;rmulas de suma. Para el seno y el coseno, son:<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>Para la tangente, la identidad de &#225;ngulo doble es:<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>Estas identidades son especialmente &#250;tiles para reducir potencias superiores de funciones trigonom&#233;tricas y para simplificar integrales.Las identidades de medio &#225;ngulo se derivan de las f&#243;rmulas de &#225;ngulo doble al despejar sen^2&#55349;&#57091; y cos^2&#55349;&#57091;. Se expresan como:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:71.81%;'><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>La elecci&#243;n del signo depende del cuadrante en el que se encuentre el medio &#225;ngulo. Estas identidades son cruciales para evaluar funciones trigonom&#233;tricas en &#225;ngulos no est&#225;ndar y en t&#233;cnicas de integraci&#243;n, como la sustituci&#243;n trigonom&#233;trica.

Identidades trigonométricas II

Las identidades trigonom&#233;tricas de &#225;ngulo doble y de medio &#225;ngulo se derivan de las f&#243;rmulas fundamentales de suma y diferencia y ...

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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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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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