27.4: Revisión de funciones trigonométricas

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Page ID: 117667

Thomas Tradler and Holly Carley
CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

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Ejercicio$\PageIndex{1}$

Rellene todos los valores de las funciones trigonométricas en la siguiente tabla.

\ [\ begin {array} {c||c|c|c|c|c}
& 0 &\ dfrac {\ pi} {6} &\ dfrac {\ pi} {4} &\ dfrac {\ pi} {3} &\ dfrac {\ pi} {2}\
\ hline\ hline\ hline\ sin (x) & & & &
\\ hline\ cos (x) & & & & &\
\ hline\ tan (x) & & amp; & & &
\ end {array}\ nonumber\]

Contestar: \ (\ begin {array} {c||c|c|c|c|c}
x & 0=0^ {\ circ} &\ dfrac {\ pi} {6} =30^ {\ circ} &\ dfrac {\ pi} {4} =45^ {\ circ} &\ dfrac {\ pi} {3} =60^ {\ circ} &\ dfrac {\ pi} {2} =90^ {\ circ}\\ hline
\ hline\ hline\ sin (x) & 0 &\ dfrac {1} {2} &\ dfrac {\ sqrt {2}} {2} &\ dfrac {\ sqrt { 3}} {2} & 1\
\ hline\ cos (x) & 1 &\ dfrac {\ sqrt {3}} {2} &\ dfrac {\ sqrt {2}} {2} &\ dfrac {1} {2} & 0\
\ hline\ tan (x) & 0 &\ dfrac {\ sqrt {3}} {3} & 1 &\ sqrt {3} &\ text {undef.}
\ end {array}\)

Ejercicio$\PageIndex{2}$

Encuentra los valores exactos de

$\cos\left(-\dfrac{\pi}{6}\right)$
$\sin\left(-\dfrac{\pi}{4}\right)$
$\tan\left(-\dfrac{\pi}{3}\right)$

Contestar

$\dfrac{\sqrt{3}}{2}$
$\dfrac{-\sqrt{2}}{2}$
$-\sqrt{3}$

Ejercicio$\PageIndex{3}$

Encuentra el valor exacto de la función trigonométrica.

$\sin\left(\dfrac{5\pi}{4}\right)$
$\cos\left(\dfrac{11\pi}{6}\right)$

[Pista: Usa el especial$45^\circ-45^\circ-90^\circ$ o$30^\circ-60^\circ-90^\circ$ triángulos para encontrar la solución.]

Contestar

$\dfrac{-\sqrt{2}}{2}$
$\dfrac{\sqrt{3}}{2}$

Ejercicio$\PageIndex{4}$

Encuentra la amplitud, el periodo y el desplazamiento de fase de la función dada. Dibuja la gráfica en un intervalo de un periodo. Etiquetar todos los máximos, mínimos e intercepciones.

$y=3 \cos\left(4 x-\pi\right)$
$y=-5\sin\left(x+\dfrac{\pi}{2}\right)$

Contestar

amplitud$3$, periodo$\dfrac{\pi}{2}$, desplazamiento de fase$\dfrac{\pi}{4}$ $clipboard_eae7381b7075b464c7074552724f5f1ed.png$
amplitud$5$, periodo$2\pi $, desplazamiento de fase$\dfrac{-\pi}{2}$ $clipboard_e24e4d7c331858767be9f4c23ff7a2c75.png$

Ejercicio$\PageIndex{5}$

Encuentra el valor exacto de la función trigonométrica.

$\cos\left(\dfrac{\pi}{12}\right)$[Pista: Utilice las fórmulas de suma y resta de ángulos.]
$\cos\left(\dfrac{3\pi}{8}\right)$[Pista: Usa las fórmulas de medio ángulos.]

Contestar

$\dfrac{\sqrt{2}+\sqrt{6}}{4}$
$\dfrac{\sqrt{2-\sqrt{2}}}{2}$

Ejercicio$\PageIndex{6}$

Dejemos$\sin(\alpha)=-\dfrac{4}{5}$ y dejemos$\alpha$ estar en el cuadrante III. Encontrar$\sin(2\alpha)$,$\cos(2\alpha)$, y$\tan(2\alpha)$.

Contestar: $\sin (2 \alpha)=\dfrac{24}{25}, \cos (2 \alpha)=\dfrac{-7}{25}, \tan (2 \alpha)=\dfrac{-24}{7}$

Ejercicio$\PageIndex{7}$

Encuentra el valor exacto de:

$\sin^{-1}\left(-\dfrac{1}{2}\right)$
$\cos^{-1}\left(-\dfrac{\sqrt{3}}{2}\right)$
$\tan^{-1}\left(-\dfrac{\sqrt{3}}{3}\right)$

Contestar

$\dfrac{-\pi}{6}$
$\dfrac{5 \pi}{6}$
$\dfrac{-\pi}{6}$

Ejercicio$\PageIndex{8}$

Resolver para$x$:$2\sin(x)+\sqrt{3}=0$

Contestar: $x=(-1)^{n+1} \dfrac{\pi}{3}+n \pi$, donde$n=0, \pm 1, \ldots $

Ejercicio$\PageIndex{9}$

Resolver para$x$:$\tan^2(x)-1=0$

Contestar: $x=\pm \dfrac{\pi}{4}+n \pi $donde$n=0, \pm 1, \ldots $

Ejercicio$\PageIndex{10}$

Resolver para$x$.

$2\cos^2(x)-1=0$
$2\sin^2(x)+15\sin(x)+7=0$

Contestar

$x=\pm \dfrac{\pi}{4}+2 n \pi$o$x=\pm \dfrac{3 \pi}{4}+2 n \pi$ donde$n=0, \pm 1, \ldots$
$(-1)^{n+1} \dfrac{\pi}{6}+n \pi$donde$n=0, \pm 1, \ldots$

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