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

Em um parque de skate, um ciclista se aproxima de uma rampa curva. Sua inclinação variável altera a aceleração do ciclista, traçando um caminho moldado pelo movimento angular.À medida que o ciclista sobe ou desce a rampa, a inclinação do caminho muda - com semelhanças com a função tangente, que compara a subida vertical com a corrida horizontal.A tangente é definida como seno dividido por cosseno para qualquer ângulo medido a partir da horizontal.A partir dessa definição, a identidade de ângulo duplo pode ser derivada. Lembre-se de seno e cosseno de duas vezes o ângulo. Substituí-los na definição de tangente leva ao resultado: tangente de duas vezes o ângulo. Quando o cavaleiro se lança mais alto e o ângulo dobra, essa identidade prevê a nova inclinação.Para ângulos menores perto da base da rampa, onde o ciclista entra em uma curva suave, a identidade de meio ângulo torna-se útil.Recordar e aplicar as identidades de meio-ângulo seno e cosseno à definição tangente dá uma expressão. Simplificá-lo tomando seu conjugado e multiplicando por um mais ou um menos cosseno produz duas formas equivalentes da identidade tangente de meio ângulo.Essas identidades auxiliam na análise de mudanças direcionais e na modelagem de movimentos curvos na dinâmica do parque de skate.

identidades trigonométricas ii

Trigonometric Identities II

Identidades trigonom&#233;tricas de &#226;ngulo duplo e de meio-&#226;ngulo s&#227;o derivadas das f&#243;rmulas fundamentais de soma e diferen&#231;a e constituem ferramentas essenciais para simplificar express&#245;es, resolver equa&#231;&#245;es e avaliar integrais. Essas identidades reduzem a complexidade das fun&#231;&#245;es trigonom&#233;tricas ao relacionar fun&#231;&#245;es de um &#226;ngulo m&#250;ltiplo ou fracion&#225;rio a fun&#231;&#245;es de um &#250;nico &#226;ngulo. Suas aplica&#231;&#245;es abrangem matem&#225;tica, f&#237;sica e engenharia, particularmente em an&#225;lise de Fourier, mec&#226;nica ondulat&#243;ria e c&#225;lculo.As identidades de &#226;ngulo duplo s&#227;o obtidas ao substituir &#226;ngulos iguais nas f&#243;rmulas de soma. Para seno e cosseno, elas s&#227;o:<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 tangente, a identidade de &#226;ngulo duplo &#233;:<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>Essas identidades s&#227;o especialmente &#250;teis para reduzir pot&#234;ncias elevadas de fun&#231;&#245;es trigonom&#233;tricas e simplificar integrais.As identidades de meio-&#226;ngulo derivam-se das f&#243;rmulas de &#226;ngulo duplo, ao resolver sen^2&#55349;&#57091; e cos^2&#55349;&#57091;. Elas s&#227;o expressas como:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:72.2%;'><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>A escolha do sinal depende do quadrante em que o meio-&#226;ngulo est&#225; localizado. Essas identidades s&#227;o fundamentais na avalia&#231;&#227;o de fun&#231;&#245;es trigonom&#233;tricas para &#226;ngulos n&#227;o convencionais e em t&#233;cnicas de integra&#231;&#227;o, como a substitui&#231;&#227;o trigonom&#233;trica.

Identidades Trigonométricas II

Identidades trigonom&#233;tricas de &#226;ngulo duplo e de meio-&#226;ngulo s&#227;o derivadas das f&#243;rmulas fundamentais de soma e diferen&#231;a e ...

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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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