Name: unit circle trig: signs and symmetry
Uploaded: 2025-11-26T05:44:36.000Z
Duration: 1 min 30 s
Channel: JoVE

El círculo unitario, centrado en el origen, define las funciones trigonométricas.Cada número real representa una longitud de arco medida desde el eje x positivo y lo gira en sentido contrario a las agujas del reloj alrededor del círculo.La coordenada x da el coseno del número real. De manera similar, la coordenada y da el seno.La tangente se deriva como seno sobre coseno, la cotangente es coseno sobre el seno, la secante es el recíproco del coseno y la cosecante es el recíproco del seno.Dividir el círculo en cuatro cuadrantes ayuda a identificar el signo de una función.Todas las funciones son positivas en el primer cuadrante, donde x e y son positivas. En el segundo, dado que x es negativo, el coseno es negativo. Como y es positivo, el seno también es positivo. La tangente, que es y sobre x, es negativa porque un positivo sobre un negativo es negativo.En el tercero, la tangente y la cotangente son positivas, ya que x e y son negativas. En el cuarto, solo el coseno y la secante son positivos, ya que x es positivo e y es negativo.Las funciones trigonométricas son útiles para cálculos del mundo real, como estimar la altura de un edificio multiplicando la distancia horizontal conocida por la tangente del ángulo de inclinación conocido.

funciones trigonométricas de números reales

Trigonometric Functions of Real Numbers

La circunferencia unitaria (una circunferencia de radio uno, centrada en el origen del plano de coordenadas) sirve como marco fundamental para definir las funciones trigonom&#233;tricas. En este contexto, la longitud de arco se refiere a la distancia medida a lo largo de la circunferencia entre dos puntos y proporciona una forma de representar geom&#233;tricamente los n&#250;meros reales. Cada n&#250;mero real t corresponde a una longitud de arco medida en sentido contrario a las agujas del reloj desde el eje x positivo alrededor de la circunferencia. Las coordenadas de un punto de la circunferencia unitaria (x, y) a una longitud de arco dada definen el coseno y el seno del n&#250;mero real, respectivamente, donde x = cos t y y = sen t.Otras funciones trigonom&#233;tricas se derivan directamente del seno y del coseno. La funci&#243;n tangente se define como el cociente entre el seno y el coseno:<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/40073/40073_Equation_1_1__editor_40073.svg' alt='40073_Equation_1_1_.svg' /></figure>Por su parte, la funci&#243;n cotangente es el cociente entre el coseno y el seno:<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/40073/40073_Equation_2_1__editor_40073.svg' alt='40073_Equation_2_1_.svg' /></figure>Las funciones secante y cosecante son las rec&#237;procas del coseno y del seno, respectivamente:<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/40073/40073_Equation_3_1__editor_40073.svg' alt='40073_Equation_3_1_.svg' /></figure>Dividir la circunferencia unitaria en cuatro cuadrantes aclara los signos de las funciones trigonom&#233;tricas. En el primer cuadrante, todas las funciones son positivas. En el segundo, solo el seno y la cosecante conservan valores positivos. El tercer cuadrante hace que la tangente y la cotangente sean positivas, mientras que el coseno y la secante lo son en el cuarto. Estas reglas de signo son esenciales para evaluar funciones trigonom&#233;tricas en distintos &#225;ngulos.La simetr&#237;a de la circunferencia unitaria confiere propiedades de paridad a las funciones. El coseno y la secante son funciones pares, que cumplen cos (&#8722;t) = cos t y sec (&#8722;t) = sec t. En cambio, el seno, la tangente, la cotangente y la cosecante son funciones impares, que satisfacen identidades como sen (&#8722;t) = &#8722;sen t y tan (&#8722;t) = &#8722;tan t.En conjunto, estas definiciones y convenciones de signo ofrecen un marco coherente para comprender el comportamiento de las funciones trigonom&#233;tricas. La circunferencia unitaria no solo proporciona una intuici&#243;n geom&#233;trica, sino que tambi&#233;n facilita el c&#225;lculo y el an&#225;lisis precisos, lo que la hace indispensable tanto en matem&#225;ticas puras como aplicadas.

Funciones trigonométricas de números reales

La circunferencia unitaria (una circunferencia de radio uno, centrada en el origen del plano de coordenadas) sirve como marco fundamental para definir las ...

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unit circle trig: signs and symmetry

The unit circle is the basis for trigonometric functions on real numbers. It is a circle with radius one centered at the origin of the coordinate plane. On this circle, arc length means the distance measured along the circumference between two points. Each real number t is shown as an arc length measured counterclockwise from the positive x-axis.

A point on the unit circle has coordinates (x, y). These coordinates define cosine and sine. The x-coordinate equals cos t, and the y-coordinate equals sin t. The other trig functions come from these two values. Tangent is the ratio of sine to cosine. Cotangent is the ratio of cosine to sine. Secant is the reciprocal of cosine, and cosecant is the reciprocal of sine.

The four quadrants of the unit circle help show where each trigonometric function is positive or negative. In the first quadrant, all trigonometric functions are positive. In the second quadrant, only sine and cosecant are positive. In the third quadrant, tangent and cotangent are positive. In the fourth quadrant, cosine and secant are positive. These sign patterns are important when evaluating trig values for different angles.

The unit circle also shows symmetry in trigonometric functions. Cosine and secant are even functions, so cos(-t) = cos t and sec(-t) = sec t. Sine, tangent, cotangent, and cosecant are odd functions. For example, sin(-t) = -sin t and tan(-t) = -tan t. Together, these rules give a clear way to study trig functions and compute them across the coordinate plane.

Unit Circle Trig: Signs and Symmetry

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trigonometric functions of real numbers

The unit circle is the basis for trigonometric functions on real numbers. It is a circle with radius one centered at the origin of the coordinate plane. On this circle, arc length means the distance measured along the circumference between two points. Each real number t is shown as an arc length measured counterclockwise from the positive x-axis. A point on the unit circle has coordinates (x, y). These coordinates define cosine and sine. The x-coordinate equals cos t, and the y-coordinate...

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Explore how the unit circle defines sine, cosine, tangent, and related functions, plus quadrant signs and even-odd symmetry for real numbers.

Unit Circle Trig: Signs and Symmetry | 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: Signs and Symmetry

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