20.3: Ejercicios
( \newcommand{\kernel}{\mathrm{null}\,}\)
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 tomarn=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
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
Encontrar al menos soluciones5 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}
Resolver parax. 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 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. ),
- 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
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