Name: trigonometric wave graphs and asymptotes
Uploaded: 2026-02-12T10:41:06.000Z
Duration: 1 min 30 s
Channel: JoVE

Los gráficos trigonométricos están relacionados con el círculo unitario. A medida que un punto se mueve en sentido contrario a las agujas del reloj alrededor del círculo, su proyección vertical traza la onda sinusoidal.De manera similar, la proyección horizontal del movimiento del punto alrededor del círculo produce la onda de coseno.Los gráficos de seno y coseno tienen un período de 2π, un dominio de todos los números reales, y un rango de -1 a 1.En el círculo unitario, la intersección de una línea tangente vertical y una secante en el círculo que pasa por el origen traza el gráfico de función tangente. La función tangente se define en todas partes excepto donde el coseno es cero. Su rango son todos números reales.El gráfico cosecante se repite cada 2π. Es indefinido donde el seno es cero. Sus valores son siempre menores que -1 o mayores que 1.El gráfico secante también tiene un período de 2π. No está definido donde el coseno es cero y tiene el mismo rango que la cosecante.La gráfica cotangente no está definida donde el seno es cero y tiene un rango de todos los números reales.Los gráficos trigonométricos modelan escenarios del mundo real, como un punto en una rueda giratoria que traza una onda sinusoidal para mostrar una posición cambiante.

gráficas de las funciones trigonométricas

Las funciones trigonom&#233;tricas presentan un comportamiento peri&#243;dico y sim&#233;trico, arraigado en la circunferencia unitaria. Las funciones seno y coseno corresponden a las proyecciones vertical y horizontal, respectivamente, de un punto que gira en sentido antihorario alrededor de la circunferencia. Estas funciones trazan formas de onda suaves y repetitivas con periodos id&#233;nticos y rangos acotados. La funci&#243;n tangente se define como el cociente entre el seno y el coseno y genera una curva no acotada que se repite cada &#960; unidades, con as&#237;ntotas verticales donde el coseno es cero. De manera an&#225;loga, la funci&#243;n cotangente es el cociente entre el coseno y el seno y se comporta de manera similar, con as&#237;ntotas donde el seno es cero.
Las funciones secante y cosecante, rec&#237;procas del coseno y del seno, respectivamente, forman curvas que se arquean alej&#225;ndose del eje x y quedan indefinidas en los ceros de sus funciones base. Estas gr&#225;ficas muestran as&#237;ntotas verticales y no intersectan el eje x. A pesar de sus diferencias, todas estas funciones exhiben periodicidad regular y simetr&#237;as caracter&#237;sticas: el seno, la tangente y la cosecante son funciones impares, sim&#233;tricas respecto del origen, mientras que el coseno y la secante son funciones pares, sim&#233;tricas respecto del eje y. Sus restricciones de dominio surgen de la divisi&#243;n por cero en sus definiciones, y sus rangos reflejan si est&#225;n acotadas o no.
Propiedades gr&#225;ficas de las funciones trigonom&#233;tricas
<table>
<tbody><tr>
<td>Funci&#243;n
</td>
<td>Periodo
</td>
<td>Dominio
</td>
<td>Rango
</td>
</tr>
<tr>
<td>Seno
</td>
<td>2&#960;
</td>
<td>Todos los n&#250;meros reales
</td>
<td>[-1, 1]
</td>
</tr>
<tr>
<td>Coseno
</td>
<td>2&#960;
</td>
<td>Todos los n&#250;meros reales
</td>
<td>[-1, 1]
</td>
</tr>
<tr>
<td>Tangente
</td>
<td>&#960; 
</td>
<td>Todos los n&#250;meros reales, excepto x = (&#960;/2) + n&#960;, donde n es cualquier entero
</td>
<td>Todos los n&#250;meros reales
</td>
</tr>
<tr>
<td>Cotangente
</td>
<td>2&#960; 
</td>
<td>Todos los n&#250;meros reales, excepto x = n&#960;, donde n es cualquier entero
</td>
<td>Todos los n&#250;meros reales
</td>
</tr>
<tr>
<td>Secante
</td>
<td>2&#960; 
</td>
<td>Todos los n&#250;meros reales, excepto x = (&#960;/2) + n&#960;, donde n es cualquier entero 
</td>
<td>Todos los n&#250;meros reales &#8804; &#8211;1 &#8746; &#8804; &#8211;1
</td>
</tr>
<tr>
<td>Cosecante
</td>
<td>2&#960; 
</td>
<td>Todos los n&#250;meros reales, excepto x = n&#960;, donde n es cualquier entero
</td>
<td>Todos los n&#250;meros reales &#8804; &#8211;1 &#8746; &#8804; &#8211;1
</td>
</tr>
</tbody></table>

Gráficas de las funciones trigonométricas

Las funciones trigonom&#233;tricas presentan un comportamiento peri&#243;dico y sim&#233;trico, arraigado en la circunferencia unitaria. Las funciones ...

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Graphs of Trigonometric Functions

trigonometric wave graphs and asymptotes

Trigonometric functions create repeating waves and curved graphs with clear symmetry. Their shapes come from the unit circle, where a point moves counterclockwise and its vertical and horizontal positions form the sine and cosine values.

Sine and cosine graphs are smooth waves with the same period, or repeat length. Their ranges stay bounded between -1 and 1. Sine, cosine, and the other trig functions all show regular patterns, but each one has its own graph shape.

Tangent is the ratio of sine to cosine, so its graph is unbounded. It repeats every period and has vertical asymptotes, which are lines the graph approaches but does not cross, where cosine equals zero. Cotangent is the ratio of cosine to sine and has similar behavior, with asymptotes where sine equals zero.

Secant and cosecant are reciprocal functions. Secant is the reciprocal of cosine, and cosecant is the reciprocal of sine. Their graphs arch away from the x-axis, become undefined at zeros of the base functions, and do not intersect the x-axis.

These trig graphs also have different symmetries. Sine, tangent, and cosecant are odd functions, so they are symmetric about the origin. Cosine and secant are even functions, so they are symmetric about the y-axis. Their domains and ranges depend on whether the function is bounded or unbounded, and the table of properties shows the period and domain restrictions for sine, cosine, tangent, cotangent, secant, and cosecant.

Trigonometric Wave Graphs and Asymptotes

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Trigonometric functions create repeating waves and curved graphs with clear symmetry. Their shapes come from the unit circle, where a point moves counterclockwise and its vertical and horizontal positions form the sine and cosine values. Sine and cosine graphs are smooth waves with the same period, or repeat length. Their ranges stay bounded between -1 and 1. Sine, cosine, and the other trig functions all show regular patterns, but each one has its own graph shape. Tangent is the ratio of...

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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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Trigonometric Wave Graphs and Asymptotes

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