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cos^2x=3/4 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   2         
cos (x) = 3/4
cos⁡2(x)=34\cos^{2}{\left(x \right)} = \frac{3}{4}
Simplificar
Solución detallada
Tenemos la ecuación
cos⁡2(x)=34\cos^{2}{\left(x \right)} = \frac{3}{4}
cambiamos
cos⁡2(x)−34=0\cos^{2}{\left(x \right)} - \frac{3}{4} = 0
cos⁡2(x)−34=0\cos^{2}{\left(x \right)} - \frac{3}{4} = 0
Sustituimos
w=cos⁡(x)w = \cos{\left(x \right)}
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:
w1=D−b2aw_{1} = \frac{\sqrt{D} - b}{2 a}
w2=−D−b2aw_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=1a = 1
b=0b = 0
c=−34c = - \frac{3}{4}
, entonces
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (1) * (-3/4) = 3

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

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

o
w1=32w_{1} = \frac{\sqrt{3}}{2}
w2=−32w_{2} = - \frac{\sqrt{3}}{2}
hacemos cambio inverso
cos⁡(x)=w\cos{\left(x \right)} = w
Tenemos la ecuación
cos⁡(x)=w\cos{\left(x \right)} = w
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
x=πn+acos⁡(w)x = \pi n + \operatorname{acos}{\left(w \right)}
x=πn+acos⁡(w)−πx = \pi n + \operatorname{acos}{\left(w \right)} - \pi
O
x=πn+acos⁡(w)x = \pi n + \operatorname{acos}{\left(w \right)}
x=πn+acos⁡(w)−πx = \pi n + \operatorname{acos}{\left(w \right)} - \pi
, donde n es cualquier número entero
sustituimos w:
x1=πn+acos⁡(w1)x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}
x1=πn+acos⁡(32)x_{1} = \pi n + \operatorname{acos}{\left(\frac{\sqrt{3}}{2} \right)}
x1=πn+π6x_{1} = \pi n + \frac{\pi}{6}
x2=πn+acos⁡(w2)x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}
x2=πn+acos⁡(−32)x_{2} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} \right)}
x2=πn+5π6x_{2} = \pi n + \frac{5 \pi}{6}
x3=πn+acos⁡(w1)−πx_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi
x3=πn−π+acos⁡(32)x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{3}}{2} \right)}
x3=πn−5π6x_{3} = \pi n - \frac{5 \pi}{6}
x4=πn+acos⁡(w2)−πx_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi
x4=πn−π+acos⁡(−32)x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} \right)}
x4=πn−π6x_{4} = \pi n - \frac{\pi}{6}
Gráfica
0-80-60-40-2020406080-10010002
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
pi   5*pi   7*pi   11*pi
-- + ---- + ---- + -----
6     6      6       6
11π6+((π6+5π6)+7π6)\frac{11 \pi}{6} + \left(\left(\frac{\pi}{6} + \frac{5 \pi}{6}\right) + \frac{7 \pi}{6}\right) 
=
4*pi
4π4 \pi 
producto
pi 5*pi 7*pi 11*pi
--*----*----*-----
6   6    6     6
11π67π6π65π6\frac{11 \pi}{6} \frac{7 \pi}{6} \frac{\pi}{6} \frac{5 \pi}{6} 
=
      4
385*pi 
-------
  1296
385π41296\frac{385 \pi^{4}}{1296} 
385*pi^4/1296
Respuesta rápida [src] 
     pi
x1 = --
     6
x1=π6x_{1} = \frac{\pi}{6} 
     5*pi
x2 = ----
      6
x2=5π6x_{2} = \frac{5 \pi}{6} 
     7*pi
x3 = ----
      6
x3=7π6x_{3} = \frac{7 \pi}{6} 
     11*pi
x4 = -----
       6
x4=11π6x_{4} = \frac{11 \pi}{6} 
x4 = 11*pi/6
Respuesta numérica [src] 
x1 = 60.2138591938044
x2 = 84.2994028713261
x3 = -217.293491873294
x4 = -313.635666583381
x5 = 30.8923277602996
x6 = -93.7241808320955
x7 = -90.5825881785057
x8 = 9.94837673636768
x9 = 90.5825881785057
x10 = 91.6297857297023
x11 = 40.317105721069
x12 = 96.8657734856853
x13 = -68.5914396033772
x14 = 69.6386371545737
x15 = 25.6563400043166
x16 = 8.90117918517108
x17 = -18.3259571459405
x18 = -82.2050077689329
x19 = 3.66519142918809
x20 = -5.75958653158129
x21 = -97.9129710368819
x22 = 56.025068989018
x23 = 74.8746249105567
x24 = -75.9218224617533
x25 = -3.66519142918809
x26 = 47.6474885794452
x27 = 5.75958653158129
x28 = -31.9395253114962
x29 = -24.60914245312
x30 = -19.3731546971371
x31 = -49.7418836818384
x32 = 0.523598775598299
x33 = -16.2315620435473
x34 = -2.61799387799149
x35 = -13.0899693899575
x36 = 34.0339204138894
x37 = -35.081117965086
x38 = 52.8834763354282
x39 = 12.0427718387609
x40 = 49.7418836818384
x41 = 27.7507351067098
x42 = 38.2227106186758
x43 = 16.2315620435473
x44 = -62.3082542961976
x45 = 2.61799387799149
x46 = 24.60914245312
x47 = -41.3643032722656
x48 = 125.140107367993
x49 = -9.94837673636768
x50 = -69.6386371545737
x51 = -27.7507351067098
x52 = 78.0162175641465
x53 = -56.025068989018
x54 = -12.0427718387609
x55 = 88.4881930761125
x56 = 53.9306738866248
x57 = -43.4586983746588
x58 = -21.4675497995303
x59 = 31.9395253114962
x60 = -46.6002910282486
x61 = 18.3259571459405
x62 = 131.423292675173
x63 = -84.2994028713261
x64 = 66.497044500984
x65 = -2686.58531759487
x66 = -71.733032256967
x67 = -60.2138591938044
x68 = 46.6002910282486
x69 = 68.5914396033772
x70 = 85.3466004225227
x71 = 63.3554518473942
x72 = -25.6563400043166
x73 = 93.7241808320955
x74 = 22.5147473507269
x75 = -34.0339204138894
x76 = 100.007366139275
x77 = -91.6297857297023
x78 = -47.6474885794452
x79 = 75.9218224617533
x80 = 82.2050077689329
x81 = 97.9129710368819
x82 = -53.9306738866248
x83 = -87.4409955249159
x84 = -81.1578102177363
x85 = -78.0162175641465
x86 = 19.3731546971371
x87 = -38.2227106186758
x88 = 44.5058959258554
x89 = -63.3554518473942
x90 = 71.733032256967
x91 = -100.007366139275
x92 = 62.3082542961976
x93 = 41.3643032722656
x94 = -85.3466004225227
x95 = -65.4498469497874
x96 = -40.317105721069
x96 = -40.317105721069
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