20.3: Ejercicios

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. $x \approx 0.443+2 n \pi, \text { or } x \approx 2.193+2 n \pi, \text { where } n=0, \pm 1, \ldots$