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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 tomar\(n=0, \pm 2, \pm 4, \ldots\)),
    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 soluciones\(5\) 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 para\(x\). 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 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\(x=\pi+2 n \pi\). ),
    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; a detailed edit history is available upon request.