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cos^2(x)=1/2

cos^2(x)=1/2 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   2         
cos (x) = 1/2
cos⁡2(x)=12\cos^{2}{\left(x \right)} = \frac{1}{2}
Simplificar
Solución detallada
Tenemos la ecuación
cos⁡2(x)=12\cos^{2}{\left(x \right)} = \frac{1}{2}
cambiamos
cos⁡(2x)2=0\frac{\cos{\left(2 x \right)}}{2} = 0
cos⁡2(x)−12=0\cos^{2}{\left(x \right)} - \frac{1}{2} = 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=−12c = - \frac{1}{2}
, entonces
D = b^2 - 4 * a * c = 

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

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

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

o
w1=22w_{1} = \frac{\sqrt{2}}{2}
w2=−22w_{2} = - \frac{\sqrt{2}}{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⁡(22)x_{1} = \pi n + \operatorname{acos}{\left(\frac{\sqrt{2}}{2} \right)}
x1=πn+π4x_{1} = \pi n + \frac{\pi}{4}
x2=πn+acos⁡(w2)x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}
x2=πn+acos⁡(−22)x_{2} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{2}}{2} \right)}
x2=πn+3π4x_{2} = \pi n + \frac{3 \pi}{4}
x3=πn+acos⁡(w1)−πx_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi
x3=πn−π+acos⁡(22)x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{2}}{2} \right)}
x3=πn−3π4x_{3} = \pi n - \frac{3 \pi}{4}
x4=πn+acos⁡(w2)−πx_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi
x4=πn−π+acos⁡(−22)x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{2}}{2} \right)}
x4=πn−π4x_{4} = \pi n - \frac{\pi}{4}
Gráfica
0-80-60-40-2020406080-10010002
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
     pi
x1 = --
     4
x1=π4x_{1} = \frac{\pi}{4} 
     3*pi
x2 = ----
      4
x2=3π4x_{2} = \frac{3 \pi}{4} 
     5*pi
x3 = ----
      4
x3=5π4x_{3} = \frac{5 \pi}{4} 
     7*pi
x4 = ----
      4
x4=7π4x_{4} = \frac{7 \pi}{4} 
x4 = 7*pi/4
Suma y producto de raíces [src] 
suma
pi   3*pi   5*pi   7*pi
-- + ---- + ---- + ----
4     4      4      4
7π4+((π4+3π4)+5π4)\frac{7 \pi}{4} + \left(\left(\frac{\pi}{4} + \frac{3 \pi}{4}\right) + \frac{5 \pi}{4}\right) 
=
4*pi
4π4 \pi 
producto
pi 3*pi 5*pi 7*pi
--*----*----*----
4   4    4    4
7π45π4π43π4\frac{7 \pi}{4} \frac{5 \pi}{4} \frac{\pi}{4} \frac{3 \pi}{4} 
=
      4
105*pi 
-------
  256
105π4256\frac{105 \pi^{4}}{256} 
105*pi^4/256
Respuesta numérica [src] 
x1 = 40.0553063332699
x2 = -3.92699081698724
x3 = -71.4712328691678
x4 = -13.3517687777566
x5 = 8.63937979737193
x6 = 99.7455667514759
x7 = -54.1924732744239
x8 = -62.0464549083984
x9 = 25.9181393921158
x10 = 38.484510006475
x11 = -91.8915851175014
x12 = 52.621676947629
x13 = 41.6261026600648
x14 = 18.0641577581413
x15 = -40.0553063332699
x16 = 63.6172512351933
x17 = -33.7721210260903
x18 = 62.0464549083984
x19 = 49.4800842940392
x20 = 2.35619449019234
x21 = 85.6083998103219
x22 = -55.7632696012188
x23 = -32.2013246992954
x24 = 22.776546738526
x25 = -49.4800842940392
x26 = 77.7544181763474
x27 = -57.3340659280137
x28 = -76.1836218495525
x29 = -47.9092879672443
x30 = -41.6261026600648
x31 = -84.037603483527
x32 = -24.3473430653209
x33 = 98.174770424681
x34 = 19.6349540849362
x35 = -69.9004365423729
x36 = 68.329640215578
x37 = -25.9181393921158
x38 = -1131.75875345572
x39 = -35.3429173528852
x40 = 76.1836218495525
x41 = 54.1924732744239
x42 = -10.2101761241668
x43 = 82.4668071567321
x44 = 24.3473430653209
x45 = -90.3207887907066
x46 = 55.7632696012188
x47 = 47.9092879672443
x48 = 384.059701901352
x49 = 74.6128255227576
x50 = 88.7499924639117
x51 = -93.4623814442964
x52 = 66.7588438887831
x53 = -5.49778714378214
x54 = -85.6083998103219
x55 = 44.7676953136546
x56 = -98.174770424681
x57 = -38.484510006475
x58 = -60.4756585816035
x59 = -63.6172512351933
x60 = 69.9004365423729
x61 = 10.2101761241668
x62 = -77.7544181763474
x63 = 32.2013246992954
x64 = 46.3384916404494
x65 = 5.49778714378214
x66 = -19.6349540849362
x67 = -2.35619449019234
x68 = -11.7809724509617
x69 = 16.4933614313464
x70 = 27.4889357189107
x71 = -16.4933614313464
x72 = 14247.9080821931
x73 = -82.4668071567321
x74 = 33.7721210260903
x75 = -46.3384916404494
x76 = 87.1791961371168
x77 = -79.3252145031423
x78 = -27.4889357189107
x79 = -68.329640215578
x80 = 30.6305283725005
x81 = 84.037603483527
x82 = 96.6039740978861
x83 = -18.0641577581413
x84 = 91.8915851175014
x85 = -99.7455667514759
x86 = 60.4756585816035
x87 = 162.577419823272
x88 = 90.3207887907066
x89 = 3.92699081698724
x90 = -12461.9126586273
x91 = 11.7809724509617
x91 = 11.7809724509617
Gráfico
cos^2(x)=1/2 la ecuación
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