Name: unit circle trig identities and sum formulas
Uploaded: 2025-11-26T05:44:29.000Z
Duration: 1 min 27 s
Channel: JoVE

Una identidad trigonométrica es una ecuación que involucra funciones trigonométricas que es verdadera para todos los ángulos.Una de esas identidades, llamada identidad pitagórica, se puede derivar usando un círculo unitario.Cualquier punto de este círculo corresponde a un ángulo medido desde el eje x positivo.La coordenada x del punto es el coseno del ángulo y la coordenada y es el seno del ángulo.Las coordenadas x e y forman los lados perpendiculares de un triángulo rectángulo, con el radio como hipotenusa.Según el teorema de Pitágoras, el cuadrado de la coordenada y más el cuadrado de la coordenada x es igual al cuadrado del radio.Dado que el radio es uno, esto demuestra que el seno cuadrado más el coseno al cuadrado es igual a uno.Esto se conoce como la identidad pitagórica y se aplica a todos los ángulos reales.Esta identidad, derivada en dos dimensiones, tiene aplicaciones que se extienden a los sistemas tridimensionales. En astronomía, las coordenadas celestes describen la posición de las estrellas en la esfera celeste.La identidad asegura que estas posiciones permanezcan consistentes en la esfera, lo que permite a los astrónomos rastrear estrellas a medida que parecen moverse por el cielo.

identidades trigonométricas i

Las identidades trigonom&#233;tricas son ecuaciones que relacionan funciones trigonom&#233;tricas y se cumplen para todos los &#225;ngulos dentro de sus dominios. Una identidad fundamental entre ellas es la identidad pitag&#243;rica, que surge directamente de la geometr&#237;a de la circunferencia unitaria. Para cualquier &#225;ngulo &#952;, un punto de la circunferencia unitaria tiene coordenadas (cos &#952;, sen &#952;) y, dado que el radio de la circunferencia es uno, el Teorema de Pit&#225;goras proporciona:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_1_2__editor_40078.svg' alt='40078_Equation_1_2_.svg' /></figure>Esta identidad sirve de base para derivar otras identidades. Al dividir la identidad pitag&#243;rica por cos ^2&#952; (suponiendo que cos &#952; &#8800; 0), se obtiene:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/450;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_2_2__editor_40078.svg' alt='40078_Equation_2_2_.svg' /></figure>La identidad pitag&#243;rica tambi&#233;n interviene en identidades m&#225;s complejas, como la f&#243;rmula de suma del coseno. Un m&#233;todo para derivarla recurre a argumentos geom&#233;tricos con puntos en la circunferencia unitaria y a la f&#243;rmula de la distancia. En esta derivaci&#243;n, las coordenadas de puntos clave implican combinaciones de valores de seno y coseno. Al aplicar la f&#243;rmula de la distancia y simplificar, la identidad pitag&#243;rica reduce expresiones que incluyen cuadrados de t&#233;rminos trigonom&#233;tricos. Esto conduce finalmente a:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_3_2__editor_40078.svg' alt='40078_Equation_3_2_.svg' /></figure>Sustituyendo t por &#8722;t en la f&#243;rmula de suma y usando las propiedades de paridad del coseno y del seno, se obtiene la f&#243;rmula de diferencia:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_4_2__editor_40078.svg' alt='40078_Equation_4_2_.svg' /></figure>Las f&#243;rmulas de suma y diferencia del seno tambi&#233;n pueden derivarse mediante identidades de cofunciones y transformaciones angulares:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_5_2__editor_40078.svg' alt='40078_Equation_5_2_.svg' /></figure>Por &#250;ltimo, las identidades de suma y diferencia de la tangente se deducen de las f&#243;rmulas de suma del seno y del coseno tomando su cociente:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/450;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_6_editor_40078.svg' alt='40078_Equation_6.svg' /></figure>Estas identidades son herramientas esenciales para la simplificaci&#243;n algebraica, la resoluci&#243;n de ecuaciones trigonom&#233;tricas y la aplicaci&#243;n del c&#225;lculo.

Identidades trigonométricas I

Las identidades trigonom&#233;tricas son ecuaciones que relacionan funciones trigonom&#233;tricas y se cumplen para todos los &#225;ngulos dentro de sus ...

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Trigonometric Identities I

unit circle trig identities and sum formulas

Trigonometric identities link trigonometric functions and stay true for every angle in their domains. A key example is the Pythagorean identity, which comes directly from the geometry of the unit circle. For an angle θ, a point on the unit circle has coordinates (cos θ, sin θ). Because the radius is 1, the Pythagorean Theorem gives the basic identity.

That same identity can be rearranged to create another useful form. If it is divided by cos²θ, assuming cos θ is not 0, it leads to a second Pythagorean identity. This result helps build later trig formulas and shows how one identity can generate others.

The Pythagorean identity also supports the cosine addition formula. One way to derive it uses points on the unit circle and the distance formula. The coordinates of the points involve sine and cosine terms. After simplifying the distance expression, the Pythagorean identity removes square terms and leads to the cosine sum formula.

The subtraction formula for cosine follows from the addition formula. Replacing t with -t and using the even and odd properties of cosine and sine gives the cosine difference identity. The sine addition and subtraction formulas can also be derived through cofunction identities and angle transformations.

The tangent addition and subtraction identities come from the sine and cosine sum formulas. Taking the ratio of the sine and cosine results produces the tangent formulas. These identities are useful for simplifying algebra, solving trigonometric equations, and working in calculus.

Unit Circle Trig Identities and Sum Formulas

unit-circle-trig-identities-and-sum-formulas

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Using Cosines to Find Triangle Sides

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trigonometric identities i

Trigonometric identities link trigonometric functions and stay true for every angle in their domains. A key example is the Pythagorean identity, which comes directly from the geometry of the unit circle. For an angle θ, a point on the unit circle has coordinates (cos θ, sin θ). Because the radius is 1, the Pythagorean Theorem gives the basic identity. That same identity can be rearranged to create another useful form. If it is divided by cos²θ, assuming cos θ is not 0, it leads to a second...

trigonometric identities ii

Double-Angle and Half-Angle Formulas

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key terms and concepts

Learn how the unit circle leads to Pythagorean, sine, cosine, and tangent identities, including addition and subtraction formulas for high school math.

Unit Circle Trig Identities and Sum 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.

math fundamentals

Unit Circle Trig Identities and Sum Formulas

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