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20.3: Ejercicios

Última actualización

30 oct 2022
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- 20.2: Ecuaciones que involucran funciones trigonométricas
- 21: Números Complejos

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 $\PageIndex{1}$

Encuentre todas las soluciones de la ecuación, y simplifique lo más posible. No aproximar la solución.

$\tan(x)=\dfrac{\sqrt{3}}{3}$
$\sin(x)=\dfrac{\sqrt{3}}{2}$
$\sin(x)=-\dfrac{\sqrt{2}}{2}$
$\cos(x)=\dfrac{\sqrt{3}}{2}$
$\cos(x)=0$
$\cos(x)=-0.5$
$\cos(x)=1$
$\sin(x)=5$
$\sin(x)=0$
$\sin(x)=-1$
$\tan(x)=-\sqrt{3}$
$\cos(x)=0.2$

Responder

$x=\dfrac{\pi}{6}+n \pi \text {, where } n=0, \pm 1, \ldots$
$x=(-1)^{n} \dfrac{\pi}{3}+n \pi \text {, where } n=0, \pm 1, \ldots$
$x=(-1)^{n+1} \dfrac{\pi}{4}+n \pi, \text { where } n=0, \pm 1, \ldots$
$x=\pm \dfrac{\pi}{6}+2 n \pi, \text { where } n=0, \pm 1, \ldots$
$x=\pm \dfrac{\pi}{2}+2 n \pi, \text { where } n=0, \pm 1, \ldots$
$x=\pm \dfrac{2 \pi}{3}+2 n \pi, \text { where } n=0, \pm 1, \ldots$
$x=2 n \pi, \text { where } n=0, \pm 1, \ldots$
no hay solución (ya que $-1 \leq \sin (x) \leq$ ),
$x=n \pi, \text { where } n=0, \pm 1, \ldots$
$x=(-1)^{n+1} \dfrac{\pi}{2}+n \pi, \text { where } n=0, \pm 1, \ldots$ (ya que cada solución aparece dos veces, es suficiente tomar $n=0, \pm 2, \pm 4, \ldots$ ),
$x=\dfrac{-\pi}{3}+n \pi, \text { where } n=0, \pm 1, \ldots$
$x=\pm \cos ^{-1}(0.2)+2 n \pi, \text { where } n=0, \pm 1, \ldots$

Ejercicio $\PageIndex{2}$

Encuentra todas las soluciones de la ecuación. Aproxime su solución con la calculadora.

$\tan(x)=6.2$
$\cos(x)=0.45$
$\sin(x)=0.91$
$\cos(x)=-.772$
$\tan(x)=-0.2$
$\sin(x)=-0.06$

Responder

$x \approx 1.411+n \pi, \text { where } n=0, \pm 1, \ldots$
$x \approx \pm 1.104+2 n \pi, \text { where } n=0, \pm 1, \ldots$
$x \approx(-1)^{n} 1.143+n \pi, \text { where } n=0, \pm 1, \ldots$
$x \approx \pm 2.453+2 n \pi, \text { where } n=0, \pm 1, \ldots$
$x \approx-0.197+n \pi, \text { where } n=0, \pm 1, \ldots$
$x \approx(-1)^{n+1} 0.06+n \pi, \text { where } n=0, \pm 1, \ldots$

Ejercicio $\PageIndex{3}$

Encontrar al menos soluciones $5$ distintas de la ecuación.

$\tan(x)=-1$
$\cos(x)=\dfrac{\sqrt{2}}{2}$
$\sin(x)=-\dfrac{\sqrt{3}}{2}$
$\tan(x)=0$
$\cos(x)=0$
$\cos(x)=0.3$
$\sin(x)=0.4$
$\sin(x)=-1$

Responder

$\dfrac{-\pi}{4}, \dfrac{3 \pi}{4}, \dfrac{7 \pi}{4}, \dfrac{-5 \pi}{4}, \dfrac{-9 \pi}{4}$
$\dfrac{\pi}{4}, \dfrac{-\pi}{4}, \dfrac{9 \pi}{4}, \dfrac{-9 \pi}{4}, \dfrac{17 \pi}{4}, \dfrac{-17 \pi}{4}$
$\dfrac{-\pi}{3}, \dfrac{4 \pi}{3}, \dfrac{5 \pi}{3}, \dfrac{-2 \pi}{3}, \dfrac{-7 \pi}{3}$
$0, \pi, 2 \pi,-\pi,-2 \pi,$
$\dfrac{\pi}{2}, \dfrac{-\pi}{2}, \dfrac{3 \pi}{2}, \dfrac{-3 \pi}{2}, \dfrac{5 \pi}{2}, \dfrac{-5 \pi}{2}$
$\cos ^{-1}(0.3),-\cos ^{-1}(0.3), \cos ^{-1}(0.3)+2 \pi,-\cos ^{-1}(0.3)+2 \pi, \cos ^{-1}(0.3)-2 \pi,-\cos ^{-1}(0.3)-2 \pi$
$\sin ^{-1}(0.4),-\sin ^{-1}(0.4)+\pi, -\sin ^{-1}(0.4)-\pi, \sin ^{-1}(0.4)+2 \pi, \sin ^{-1}(0.4)-2 \pi$
$\dfrac{3 \pi}{2}, \dfrac{7 \pi}{2}, \dfrac{11 \pi}{2}, \dfrac{-\pi}{2}, \dfrac{-5 \pi}{2}$

Ejercicio $\PageIndex{4}$

Resolver para $x$ . Indicar la solución general sin aproximación.

$\tan(x)-1=0$
$2\sin(x)=1$
$2\cos(x)-\sqrt{3}=0$
$\sec(x)=-2$
$\cot(x)=\sqrt{3}$
$\tan^2(x)-3=0$
$\sin^2(x)-1=0$
$\cos^2(x)+7\cos(x)+6=0$
$4\cos^2(x)-4\cos(x)+1=0$
$2\sin^2(x)+11\sin(x)=-5$
$2\sin^2(x)+\sin(x)-1=0$
$2\cos^2(x)-3\cos(x)+1=0$
$2\cos^2(x)+9\cos(x)=5$
$\tan^3(x)-\tan(x)=0$

Responder

$x=\dfrac{\pi}{4}+n \pi, \text { where } n=0, \pm 1, \ldots$
$x=(-1)^{n} \dfrac{\pi}{6}+n \pi, \text { where } n=0, \pm 1, \ldots$
$x=\pm \dfrac{\pi}{6}+2 n \pi, \text { where } n=0, \pm 1, \ldots$
$x=\pm \dfrac{2 \pi}{3}+2 n \pi \text { where } n=0, \pm 1, \ldots$
$x=\dfrac{\pi}{6}+n \pi, \text { where } n=0, \pm 1, \ldots$
$x=\pm \dfrac{\pi}{3}+n \pi, \text { where } n=0, \pm 1, \ldots$
$x=\pm \dfrac{\pi}{2}+n \pi, \text { where } n=0, \pm 1, \ldots$
$x=\pi+2 n \pi \text {, where } n=0, \pm 1, \ldots$ (Nota: La solución que da la fórmula 20.1.5 es $x=\pm \pi+2 n \pi \text { with } n=0, \pm 1, \ldots$ Dado que cada solución aparece dos veces en esta expresión, podemos reducirla a $x=\pi+2 n \pi$ . ),
$x=\pm \dfrac{\pi}{3}+2 n \pi, \text { where } n=0, \pm 1, \ldots$
$x=(-1)^{n+1} \dfrac{\pi}{6}+n \pi, \text { where } n=0, \pm 1, \ldots$
$x=(-1)^{n+1} \dfrac{\pi}{2}+n \pi$
$x=2 n \pi, \text { or } x=\pm \dfrac{\pi}{3}+2 n \pi, \text { where } n=0, \pm 1, \ldots$
$x=\pm \dfrac{\pi}{3}+n \pi, \text { where } n=0, \pm 1, \ldots$
$x=\pm \dfrac{\pi}{4}+n \pi, \text { or } x=n \pi, \text { where } n=0, \pm 1, \ldots$

Ejercicio $\PageIndex{5}$

Utilice la calculadora para encontrar todas las soluciones de la ecuación dada. Aproximar la respuesta a la milésima más cercana.

$2\cos(x)=2\sin(x)+1$
$7\tan(x)\cdot \cos(2x)=1$
$4\cos^2(3x)+\cos(3x)=\sin(3x)+2$
$\sin(x)+\tan(x)=\cos(x)$

Responder

$x \approx-1.995+2 n \pi, \text { or } x \approx 0.424+2 n \pi, \text { where } n=0, \pm 1, \ldots$
$x \approx-0.848+n \pi, \text { or } x \approx 0.148+n \pi, \text { or } x \approx 0.700+n \pi, \text { where } n=0, \pm 1, \ldots$
$x \approx 0.262+n \dfrac{2 \pi}{3} \text {, or } x \approx 0.906+n \dfrac{2 \pi}{3} \text {, or } x \approx 1.309+n \dfrac{2 \pi}{3}, \text { or } x \approx 1.712+n \dfrac{2 \pi}{3}, \text { where } n=0, \pm 1, \ldots$
$x \approx 0.443+2 n \pi, \text { or } x \approx 2.193+2 n \pi, \text { where } n=0, \pm 1, \ldots$

Ejercicio20.3.1\PageIndex{1}

Ejercicio20.3.2\PageIndex{2}

Ejercicio20.3.3\PageIndex{3}

Ejercicio20.3.4\PageIndex{4}

Ejercicio20.3.5\PageIndex{5}

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Ejercicio $\PageIndex{1}$

Ejercicio $\PageIndex{2}$

Ejercicio $\PageIndex{3}$

Ejercicio $\PageIndex{4}$

Ejercicio $\PageIndex{5}$