17.3: Ejercicios

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Page ID: 117826

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$\PageIndex{1}$

Encuentra$\sin(x)$,$\cos(x)$, y$\tan(x)$ para los siguientes ángulos.

$x=120^\circ$
$x=390^\circ$
$x=-150^\circ$
$x=-45^\circ$
$x=1050^\circ$
$x=-810^\circ$
$x=\dfrac{5\pi}{4}$
$x=\dfrac{5\pi}{6}$
$x=\dfrac{10\pi}{3}$
$x=\dfrac{15\pi}{2}$
$x=\dfrac{-\pi}{6}$
$x=\dfrac{-54\pi}{8}$

Contestar

$\sin \left(120^{\circ}\right)=\dfrac{\sqrt{3}}{2}, \cos \left(120^{\circ}\right)=-\dfrac{1}{2}, \tan \left(120^{\circ}\right)=-\sqrt{3}$
$\sin \left(390^{\circ}\right)=\dfrac{1}{2}, \cos \left(390^{\circ}\right)=\dfrac{\sqrt{3}}{2}, \tan \left(390^{\circ}\right)=\dfrac{\sqrt{3}}{3}$
$\sin \left(-150^{\circ}\right)=-\dfrac{1}{2}, \cos \left(-150^{\circ}\right)=-\dfrac{\sqrt{3}}{2}, \tan \left(-150^{\circ}\right)=\dfrac{\sqrt{3}}{3}$
$\sin \left(-45^{\circ}\right)=-\dfrac{\sqrt{2}}{2}, \cos \left(-45^{\circ}\right)=\dfrac{\sqrt{2}}{2}, \tan \left(-45^{\circ}\right)=-1$
$\sin \left(1050^{\circ}\right)=-\dfrac{1}{2}, \cos \left(1050^{\circ}\right)=\dfrac{\sqrt{3}}{2}, \tan \left(1050^{\circ}\right)=-\dfrac{\sqrt{3}}{3}$
$\sin \left(-810^{\circ}\right)=-1, \cos \left(-810^{\circ}\right)=0, \tan \left(-810^{\circ}\right)$está indefinido
$\sin \left(\dfrac{5 \pi}{4}\right)=-\dfrac{\sqrt{2}}{2}, \cos \left(\dfrac{5 \pi}{4}\right)=-\dfrac{\sqrt{2}}{2}, \tan \left(\dfrac{5 \pi}{4}\right)=1$
$\sin \left(\dfrac{5 \pi}{6}\right)=\dfrac{1}{2}, \cos \left(\dfrac{5 \pi}{6}\right)=-\dfrac{\sqrt{3}}{2}, \tan \left(\dfrac{5 \pi}{6}\right)=-\dfrac{\sqrt{3}}{3}$
$\sin \left(\dfrac{10 \pi}{3}\right)=-\dfrac{\sqrt{3}}{2}, \cos \left(\dfrac{10 \pi}{3}\right)=-\dfrac{1}{2}, \tan \left(\dfrac{10 \pi}{3}\right)=\sqrt{3}$
$\sin \left(\dfrac{15 \pi}{2}\right)=-1, \cos \left(\dfrac{15 \pi}{2}\right)=0, \tan \left(\dfrac{15 \pi}{2}\right)$está indefinido
$\sin \left(\dfrac{-\pi}{6}\right)=-\dfrac{1}{2}, \cos \left(\dfrac{-\pi}{6}\right)=\dfrac{\sqrt{3}}{2}, \tan \left(\dfrac{-\pi}{6}\right)=-\dfrac{\sqrt{3}}{3}$
$\sin \left(\dfrac{-54 \pi}{8}\right)=-\dfrac{\sqrt{2}}{2}, \cos \left(\dfrac{-54 \pi}{8}\right)=-\dfrac{\sqrt{2}}{2}, \tan \left(\dfrac{-54 \pi}{8}\right)=1$

Ejercicio$\PageIndex{2}$

Grafique la función, y describa cómo se puede obtener la gráfica a partir de una de las gráficas básicas$y=\sin(x)$,$y=\cos(x)$, o$y=\tan(x)$.

$f(x)=\sin(x)+2$
$f(x)=\cos(x-\pi)$
$f(x)=\tan(x)-4$
$f(x)=5\cdot \sin(x)$
$f(x)=\cos(2\cdot x)$
$f(x)=\sin(x-2)-5$

Contestar

cambio hacia$y = \sin(x)$ arriba por$2$ $clipboard_e87a9fec74db13512dc88d00c5a408b50.png$
$y = \cos(x)$desplazado a la derecha$\pi$ $clipboard_ea49e58b7b0e659038bf9635b9ba96f8e.png$
$y = \tan(x)$desplazado hacia abajo$4$ $clipboard_e4568fbe75b058d26e053e360fb41df95.png$
$y = \sin(x)$estirado lejos del$x$ eje -por un factor$5$ $clipboard_ea70bf9f705034f50ed58be4f251a6281.png$
$y = \cos(x)$comprimido hacia el$y$ eje -por un factor$2$ $clipboard_e080744b55078ea167a3afba862c78862.png$
$y = \sin(x)$desplazado hacia la derecha por$2$ y hacia abajo por$5$ $clipboard_e7b1355ee1b59ad066326d55ba2459d2a.png$

Ejercicio$\PageIndex{3}$

Identificar las fórmulas con las gráficas. \ [\ begin {array} {lll}
f (x) =\ sin (x) +2, & g (x) =\ tan (x-1), & h (x) =3\ sin (x),\\
i (x) =3\ cos (x), & j (x) =\ cos (x-\ pi), & k (x) =\ tan (x) -1
\ end {array}\ nonumber\]

$clipboard_ed9f7f6deab20c9a085dbd10a02f684dd.png$
$clipboard_e020683b23064247032aac69fc1c7a107.png$
$clipboard_e5469166b94ce909f729ca288a1aed298.png$
$clipboard_e06decdf4a038417807f651176e3dd831.png$
$clipboard_e51f2385d56e4e68d886ca212744a7316.png$
$clipboard_ecf741760cde398a86b7a8ec8776b2587.png$

Contestar

$g(x)$
$h(x)$
$j(x)$
$k(x)$
$i(x)$
$f(x)$

Ejercicio$\PageIndex{4}$

Encuentra la fórmula de una función cuya gráfica es la que se muestra a continuación.

$clipboard_ea430c8007e6555ae5fafacf64920c0b9.png$
$clipboard_eef7dfc0304d8751afc660afd06d91a26.png$
$clipboard_e3cdb414f98f2d1e143a135131b7718ba.png$
$clipboard_ea748c6feb830be4ec4d0323292225119.png$
$clipboard_ec442cb9f2d0e69954a33c055884fafe1.png$
$clipboard_ecf5d99db8f8e0d021c4d07ffcfef9c6e.png$

Contestar

$y = 5 \cos(x)$
$y = −5 \cos(x)$
$y = −5 \sin(x)$
$y = \cos(x) + 5$
$y = \sin(x) + 5$
$y = 2 \sin(x) + 3$

Ejercicio$\PageIndex{5}$

Encuentra la amplitud, el periodo y el desplazamiento de fase de la función.

$f(x)=5\sin(2x+3)$
$f(x)=\sin(\pi x-5)$
$f(x)=6\sin(4x)$
$f(x)=-2\cos\left(x+\dfrac{\pi}{4}\right)$
$f(x)=8\cos(2x-6)$
$f(x)=3\sin\left(\dfrac{x}{4}\right)$
$f(x)=-\cos(x+2)$
$f(x)=7\sin \left(\dfrac{2\pi}{5}x-\dfrac{6\pi}{5}\right)$
$f(x)=\cos(-2x)$

Contestar

amplitud$5$, periodo$\pi$, desplazamiento de fase$\dfrac{−3}{2}$
amplitud$1$, periodo$2$, desplazamiento de fase$\dfrac 5 \pi$
amplitud$6$, periodo$\dfrac \pi 2$, desplazamiento de fase$0$
amplitud$2$, periodo$\dfrac 2 \pi$, desplazamiento de fase$\dfrac{−\pi}{4}$
amplitud$8$, periodo$\pi$, desplazamiento de fase$3$
amplitud$3$, periodo$\dfrac 8 \pi$, desplazamiento de fase$0$
amplitud$1$, periodo$\dfrac 2 \pi$, desplazamiento de fase$−2$
amplitud$7$, periodo$5$, desplazamiento de fase$3$
amplitud$1$, periodo$\pi$, desplazamiento de fase$0$

Ejercicio$\PageIndex{6}$

Encuentra la amplitud, el periodo y el desplazamiento de fase de la función. Utilice esta información para graficar la función a lo largo de un periodo completo. Etiquetar todos los máximos, mínimos y ceros de la función.

$y=5\cos(2x)$
$y=4\sin(\pi x)$
$y=2\sin\left(\dfrac{2\pi}{3}x\right)$
$y=\cos(2x-\pi)$
$y=\cos(\pi x-\pi)$
$y=-6\cos(-\dfrac{x}{4})$
$y=-\cos(4x+\pi)$
$y=7\sin\left(x+\dfrac{\pi}{4}\right)$
$y=5\cos\left(x+\dfrac{3\pi}{2}\right)$
$y=4\sin(5x-\pi)$
$y=-3\cos(2\pi x-4)$
$y=7\sin\left(\dfrac 1 4 x+\dfrac{\pi}{4}\right)$
$y=\cos(3x-4\pi)$
$y=2\sin\big(\dfrac 1 5 x-\dfrac{\pi}{10}\big)$
$y=\dfrac 1 3 \cos\left(\dfrac{14}{5}x-\dfrac{6\pi}{5}\right)$

Contestar

amplitud$5$, periodo$\pi$, desplazamiento de fase$0$ $clipboard_e113627ff02d07cc9bea67038b4d8a974.png$
amplitud$4$, periodo$2$, desplazamiento de fase$0$ $clipboard_edef39b269c155473be986ec93e2577c2.png$
amplitud$2$, periodo$3$, desplazamiento de fase$0$ $clipboard_e6ab47a8cab9e849a888344707a04ecda.png$
amplitud$1$, periodo$\pi$, desplazamiento de fase$\dfrac \pi 2$ $clipboard_e64231c2bd27d118a37761a9bddfafb9d.png$
amplitud$1$, periodo$2$, desplazamiento de fase$1$ $clipboard_e7ec6ef7ac4ef9917dd623cbfe2a5ef3d.png$
amplitud$6$, periodo$\dfrac 8 \pi $, desplazamiento de fase$0$ $clipboard_ed1774e04d9474194870114a705c9b522.png$
amplitud$1$, periodo$\dfrac \pi 2$, desplazamiento de fase$\dfrac {−\pi}{4}$ $clipboard_e3eb28effa7317eab90f2b0de9a48367d.png$
amplitud$7$, periodo$\dfrac 2 \pi$, desplazamiento de fase$\dfrac {−\pi}{4}$ $clipboard_ec0268bfd6521121da67ce54e9eba246b.png$
amplitud$5$, periodo$\dfrac 2 \pi$, desplazamiento de fase$\dfrac {−3\pi}{2}$ $clipboard_e41db0af7dfe18c464fc5480985a46120.png$
amplitud$4$, periodo$\dfrac {2\pi}{5}$, desplazamiento de fase$\dfrac \pi 5$ $clipboard_e1fcc37d96d30151b0cf20a260141868c.png$
amplitud$3$, periodo$1$, desplazamiento de fase$\dfrac 2 \pi$ $clipboard_e6675a6425361cf8d163fd4982cde4056.png$
amplitud$7$, periodo$\dfrac 8 \pi$, desplazamiento de fase$-\pi$ $clipboard_ea018243cb0d56c545263af605094776f.png$
amplitud$1$, periodo$\dfrac {2\pi}{3}$, desplazamiento de fase$\dfrac {4\pi}{3}$ $clipboard_e47af696f68a6368c441fb4b5a6f6c833.png$
amplitud$2$, periodo$\dfrac 10 \pi$, desplazamiento de fase$\dfrac \pi 2$ $clipboard_e32bbf0c4182ae4dc97ce0163f8be391b.png$
amplitud$\dfrac 1 3$, periodo$\dfrac {5\pi}{7}$, desplazamiento de fase$\dfrac {3\pi}{7}$ $clipboard_ef80d92a6baf65f7530888a753667da1c.png$