Sr Examen

Otras calculadoras

2*cos(x)^(2)+(sqrt(3)+2)*cos(x)+sqrt(3)=0 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

v

Solución numérica:

Buscar la solución numérica en el intervalo [, ]

Solución

Ha introducido [src]
     2      /  ___    \            ___    
2*cos (x) + \\/ 3  + 2/*cos(x) + \/ 3  = 0
$$\left(2 \cos^{2}{\left(x \right)} + \left(\sqrt{3} + 2\right) \cos{\left(x \right)}\right) + \sqrt{3} = 0$$
Solución detallada
Tenemos la ecuación
$$\left(2 \cos^{2}{\left(x \right)} + \left(\sqrt{3} + 2\right) \cos{\left(x \right)}\right) + \sqrt{3} = 0$$
cambiamos
$$2 \cos^{2}{\left(x \right)} + \left(\sqrt{3} + 2\right) \cos{\left(x \right)} + \sqrt{3} = 0$$
$$\left(2 \cos^{2}{\left(x \right)} + \left(\sqrt{3} + 2\right) \cos{\left(x \right)}\right) + \sqrt{3} = 0$$
Sustituimos
$$w = \cos{\left(x \right)}$$
Abramos la expresión en la ecuación
$$2 w^{2} + w \left(\sqrt{3} + 2\right) + \sqrt{3} = 0$$
Obtenemos la ecuación cuadrática
$$2 w^{2} + \sqrt{3} w + 2 w + \sqrt{3} = 0$$
Es la ecuación de la forma
a*w^2 + b*w + c = 0

La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 2$$
$$b = \sqrt{3} + 2$$
$$c = \sqrt{3}$$
, entonces
D = b^2 - 4 * a * c = 

(2 + sqrt(3))^2 - 4 * (2) * (sqrt(3)) = (2 + sqrt(3))^2 - 8*sqrt(3)

Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)

w2 = (-b - sqrt(D)) / (2*a)

o
$$w_{1} = - \frac{1}{2} - \frac{\sqrt{3}}{4} + \frac{\sqrt{- 8 \sqrt{3} + \left(\sqrt{3} + 2\right)^{2}}}{4}$$
$$w_{2} = - \frac{1}{2} - \frac{\sqrt{3}}{4} - \frac{\sqrt{- 8 \sqrt{3} + \left(\sqrt{3} + 2\right)^{2}}}{4}$$
hacemos cambio inverso
$$\cos{\left(x \right)} = w$$
Tenemos la ecuación
$$\cos{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
O
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} - \frac{\sqrt{3}}{4} + \frac{\sqrt{- 8 \sqrt{3} + \left(\sqrt{3} + 2\right)^{2}}}{4} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} - \frac{\sqrt{3}}{4} + \frac{\sqrt{- 8 \sqrt{3} + \left(\sqrt{3} + 2\right)^{2}}}{4} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} - \frac{\sqrt{3}}{4} - \frac{\sqrt{- 8 \sqrt{3} + \left(\sqrt{3} + 2\right)^{2}}}{4} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} - \frac{\sqrt{3}}{4} - \frac{\sqrt{- 8 \sqrt{3} + \left(\sqrt{3} + 2\right)^{2}}}{4} \right)}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} - \frac{\sqrt{3}}{4} + \frac{\sqrt{- 8 \sqrt{3} + \left(\sqrt{3} + 2\right)^{2}}}{4} \right)}$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} - \frac{\sqrt{3}}{4} + \frac{\sqrt{- 8 \sqrt{3} + \left(\sqrt{3} + 2\right)^{2}}}{4} \right)}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} - \frac{\sqrt{3}}{4} - \frac{\sqrt{- 8 \sqrt{3} + \left(\sqrt{3} + 2\right)^{2}}}{4} \right)}$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} - \frac{\sqrt{3}}{4} - \frac{\sqrt{- 8 \sqrt{3} + \left(\sqrt{3} + 2\right)^{2}}}{4} \right)}$$
Gráfica
Respuesta rápida [src]
     5*pi
x1 = ----
      6  
$$x_{1} = \frac{5 \pi}{6}$$
x2 = pi
$$x_{2} = \pi$$
     7*pi
x3 = ----
      6  
$$x_{3} = \frac{7 \pi}{6}$$
x3 = 7*pi/6
Suma y producto de raíces [src]
suma
5*pi        7*pi
---- + pi + ----
 6           6  
$$\frac{7 \pi}{6} + \left(\frac{5 \pi}{6} + \pi\right)$$
=
3*pi
$$3 \pi$$
producto
5*pi    7*pi
----*pi*----
 6       6  
$$\frac{7 \pi}{6} \frac{5 \pi}{6} \pi$$
=
     3
35*pi 
------
  36  
$$\frac{35 \pi^{3}}{36}$$
35*pi^3/36
Respuesta numérica [src]
x1 = -53.9306738866248
x2 = 41.3643032722656
x3 = 79.0634151153431
x4 = 72.7802298081635
x5 = -2.61799387799149
x6 = -59.6902604543835
x7 = -34.0339204138894
x8 = -46.6002910282486
x9 = -84.2994028713261
x10 = -85.3466004225227
x11 = -15.7079632951236
x12 = 52.8834763354282
x13 = -47.6474885794452
x14 = 34.0339204138894
x15 = 46.6002910282486
x16 = -8.90117918517108
x17 = -41.3643032722656
x18 = -40.317105721069
x19 = 2.61799387799149
x20 = -28.7979326579064
x21 = -91.6297857297023
x22 = -9.94837673636768
x23 = 97.9129710368819
x24 = 283386.841718292
x25 = -66.497044500984
x26 = 60.2138591938044
x27 = 78.0162175641465
x28 = -65.4498469497874
x29 = 53.9306738866248
x30 = -79.0634151153431
x31 = -91.1061871412841
x32 = 90.5825881785057
x33 = 16.2315620435473
x34 = -13555.9723010284
x35 = 91.6297857297023
x36 = 8.90117918517108
x37 = 21.4675497995303
x38 = 65.9734457513366
x39 = 28.7979326579064
x40 = 47.6474885794452
x41 = -52.8834763354282
x42 = 22.5147473507269
x43 = -34.5575185282487
x44 = -96.8657734856853
x45 = -90.5825881785057
x46 = 66.497044500984
x47 = -97.9129710368819
x48 = -22.5147473507269
x49 = 28.2743338656055
x50 = 96.8657734856853
x51 = -21.4675497995303
x52 = 65.4498469497874
x53 = 85.3466004225227
x54 = -27.7507351067098
x55 = -21.9911485866282
x56 = 84.8230015329624
x57 = 21.991148585022
x58 = -78.0162175641465
x59 = 15.1843644923507
x60 = -3.66519142918809
x61 = 72.2566310277441
x62 = 40.317105721069
x63 = 84.2994028713261
x64 = -71.733032256967
x65 = 59.1666616426078
x66 = 27.7507351067098
x67 = -16.2315620435473
x68 = 35.081117965086
x69 = 3.66519142918809
x70 = 9.94837673636768
x71 = -59.1666616426078
x72 = 71.733032256967
x73 = -15.1843644923507
x74 = -35.081117965086
x75 = -60.2138591938044
x76 = -47.1238895207292
x77 = -65.9734457666071
x78 = -72.7802298081635
x78 = -72.7802298081635