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20.3: Ejercicios

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Ejercicio\PageIndex{1}

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

  1. \tan(x)=\dfrac{\sqrt{3}}{3}
  2. \sin(x)=\dfrac{\sqrt{3}}{2}
  3. \sin(x)=-\dfrac{\sqrt{2}}{2}
  4. \cos(x)=\dfrac{\sqrt{3}}{2}
  5. \cos(x)=0
  6. \cos(x)=-0.5
  7. \cos(x)=1
  8. \sin(x)=5
  9. \sin(x)=0
  10. \sin(x)=-1
  11. \tan(x)=-\sqrt{3}
  12. \cos(x)=0.2
Responder
  1. x=\dfrac{\pi}{6}+n \pi \text {, where } n=0, \pm 1, \ldots
  2. x=(-1)^{n} \dfrac{\pi}{3}+n \pi \text {, where } n=0, \pm 1, \ldots
  3. x=(-1)^{n+1} \dfrac{\pi}{4}+n \pi, \text { where } n=0, \pm 1, \ldots
  4. x=\pm \dfrac{\pi}{6}+2 n \pi, \text { where } n=0, \pm 1, \ldots
  5. x=\pm \dfrac{\pi}{2}+2 n \pi, \text { where } n=0, \pm 1, \ldots
  6. x=\pm \dfrac{2 \pi}{3}+2 n \pi, \text { where } n=0, \pm 1, \ldots
  7. x=2 n \pi, \text { where } n=0, \pm 1, \ldots
  8. no hay solución (ya que-1 \leq \sin (x) \leq),
  9. x=n \pi, \text { where } n=0, \pm 1, \ldots
  10. 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 tomarn=0, \pm 2, \pm 4, \ldots),
  11. x=\dfrac{-\pi}{3}+n \pi, \text { where } n=0, \pm 1, \ldots
  12. 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.

  1. \tan(x)=6.2
  2. \cos(x)=0.45
  3. \sin(x)=0.91
  4. \cos(x)=-.772
  5. \tan(x)=-0.2
  6. \sin(x)=-0.06
Responder
  1. x \approx 1.411+n \pi, \text { where } n=0, \pm 1, \ldots
  2. x \approx \pm 1.104+2 n \pi, \text { where } n=0, \pm 1, \ldots
  3. x \approx(-1)^{n} 1.143+n \pi, \text { where } n=0, \pm 1, \ldots
  4. x \approx \pm 2.453+2 n \pi, \text { where } n=0, \pm 1, \ldots
  5. x \approx-0.197+n \pi, \text { where } n=0, \pm 1, \ldots
  6. x \approx(-1)^{n+1} 0.06+n \pi, \text { where } n=0, \pm 1, \ldots

Ejercicio\PageIndex{3}

Encontrar al menos soluciones5 distintas de la ecuación.

  1. \tan(x)=-1
  2. \cos(x)=\dfrac{\sqrt{2}}{2}
  3. \sin(x)=-\dfrac{\sqrt{3}}{2}
  4. \tan(x)=0
  5. \cos(x)=0
  6. \cos(x)=0.3
  7. \sin(x)=0.4
  8. \sin(x)=-1
Responder
  1. \dfrac{-\pi}{4}, \dfrac{3 \pi}{4}, \dfrac{7 \pi}{4}, \dfrac{-5 \pi}{4}, \dfrac{-9 \pi}{4}
  2. \dfrac{\pi}{4}, \dfrac{-\pi}{4}, \dfrac{9 \pi}{4}, \dfrac{-9 \pi}{4}, \dfrac{17 \pi}{4}, \dfrac{-17 \pi}{4}
  3. \dfrac{-\pi}{3}, \dfrac{4 \pi}{3}, \dfrac{5 \pi}{3}, \dfrac{-2 \pi}{3}, \dfrac{-7 \pi}{3}
  4. 0, \pi, 2 \pi,-\pi,-2 \pi,
  5. \dfrac{\pi}{2}, \dfrac{-\pi}{2}, \dfrac{3 \pi}{2}, \dfrac{-3 \pi}{2}, \dfrac{5 \pi}{2}, \dfrac{-5 \pi}{2}
  6. \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
  7. \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
  8. \dfrac{3 \pi}{2}, \dfrac{7 \pi}{2}, \dfrac{11 \pi}{2}, \dfrac{-\pi}{2}, \dfrac{-5 \pi}{2}

Ejercicio\PageIndex{4}

Resolver parax. Indicar la solución general sin aproximación.

  1. \tan(x)-1=0
  2. 2\sin(x)=1
  3. 2\cos(x)-\sqrt{3}=0
  4. \sec(x)=-2
  5. \cot(x)=\sqrt{3}
  6. \tan^2(x)-3=0
  7. \sin^2(x)-1=0
  8. \cos^2(x)+7\cos(x)+6=0
  9. 4\cos^2(x)-4\cos(x)+1=0
  10. 2\sin^2(x)+11\sin(x)=-5
  11. 2\sin^2(x)+\sin(x)-1=0
  12. 2\cos^2(x)-3\cos(x)+1=0
  13. 2\cos^2(x)+9\cos(x)=5
  14. \tan^3(x)-\tan(x)=0
Responder
  1. x=\dfrac{\pi}{4}+n \pi, \text { where } n=0, \pm 1, \ldots
  2. x=(-1)^{n} \dfrac{\pi}{6}+n \pi, \text { where } n=0, \pm 1, \ldots
  3. x=\pm \dfrac{\pi}{6}+2 n \pi, \text { where } n=0, \pm 1, \ldots
  4. x=\pm \dfrac{2 \pi}{3}+2 n \pi \text { where } n=0, \pm 1, \ldots
  5. x=\dfrac{\pi}{6}+n \pi, \text { where } n=0, \pm 1, \ldots
  6. x=\pm \dfrac{\pi}{3}+n \pi, \text { where } n=0, \pm 1, \ldots
  7. x=\pm \dfrac{\pi}{2}+n \pi, \text { where } n=0, \pm 1, \ldots
  8. x=\pi+2 n \pi \text {, where } n=0, \pm 1, \ldots(Nota: La solución que da la fórmula 20.1.5 esx=\pm \pi+2 n \pi \text { with } n=0, \pm 1, \ldots Dado que cada solución aparece dos veces en esta expresión, podemos reducirla ax=\pi+2 n \pi. ),
  9. x=\pm \dfrac{\pi}{3}+2 n \pi, \text { where } n=0, \pm 1, \ldots
  10. x=(-1)^{n+1} \dfrac{\pi}{6}+n \pi, \text { where } n=0, \pm 1, \ldots
  11. x=(-1)^{n+1} \dfrac{\pi}{2}+n \pi
  12. x=2 n \pi, \text { or } x=\pm \dfrac{\pi}{3}+2 n \pi, \text { where } n=0, \pm 1, \ldots
  13. x=\pm \dfrac{\pi}{3}+n \pi, \text { where } n=0, \pm 1, \ldots
  14. 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.

  1. 2\cos(x)=2\sin(x)+1
  2. 7\tan(x)\cdot \cos(2x)=1
  3. 4\cos^2(3x)+\cos(3x)=\sin(3x)+2
  4. \sin(x)+\tan(x)=\cos(x)
Responder
  1. x \approx-1.995+2 n \pi, \text { or } x \approx 0.424+2 n \pi, \text { where } n=0, \pm 1, \ldots
  2. 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
  3. 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
  4. x \approx 0.443+2 n \pi, \text { or } x \approx 2.193+2 n \pi, \text { where } n=0, \pm 1, \ldots

This page titled 20.3: Ejercicios is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform.

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