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cos(2*x)=sqrt(2)/2

cos(2*x)=sqrt(2)/2 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
             ___
           \/ 2 
cos(2*x) = -----
             2  
$$\cos{\left(2 x \right)} = \frac{\sqrt{2}}{2}$$
Solución detallada
Tenemos la ecuación
$$\cos{\left(2 x \right)} = \frac{\sqrt{2}}{2}$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$2 x = \pi n + \operatorname{acos}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$2 x = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{2}}{2} \right)}$$
O
$$2 x = \pi n + \frac{\pi}{4}$$
$$2 x = \pi n - \frac{3 \pi}{4}$$
, donde n es cualquier número entero
Dividamos ambos miembros de la ecuación obtenida en
$$2$$
obtenemos la respuesta:
$$x_{1} = \frac{\pi n}{2} + \frac{\pi}{8}$$
$$x_{2} = \frac{\pi n}{2} - \frac{3 \pi}{8}$$
Gráfica
Suma y producto de raíces [src]
suma
pi   7*pi
-- + ----
8     8  
$$\frac{\pi}{8} + \frac{7 \pi}{8}$$
=
pi
$$\pi$$
producto
pi 7*pi
--*----
8   8  
$$\frac{\pi}{8} \frac{7 \pi}{8}$$
=
    2
7*pi 
-----
  64 
$$\frac{7 \pi^{2}}{64}$$
7*pi^2/64
Respuesta rápida [src]
     pi
x1 = --
     8 
$$x_{1} = \frac{\pi}{8}$$
     7*pi
x2 = ----
      8  
$$x_{2} = \frac{7 \pi}{8}$$
x2 = 7*pi/8
Respuesta numérica [src]
x1 = 68.7223392972767
x2 = -82.0741080750334
x3 = 47.5165888855456
x4 = 88.3572933822129
x5 = 66.3661448070844
x6 = 18.45685683984
x7 = 53.7997741927252
x8 = 46.7311907221482
x9 = 5.89048622548086
x10 = 96.9966731795849
x11 = -19.2422550032375
x12 = 34.164820107789
x13 = -93.8550805259951
x14 = -78.1471172580461
x15 = 60.0829594999048
x16 = 22.3838476568273
x17 = 71.8639319508665
x18 = -49.872783375738
x19 = 9.8174770424681
x20 = -84.4303025652257
x21 = -24.7400421470196
x22 = 49.872783375738
x23 = 2.74889357189107
x24 = -286.27763055837
x25 = -38.0918109247762
x26 = 38.0918109247762
x27 = -43.5895980685584
x28 = -40.4480054149686
x29 = -85.2157007286231
x30 = -100.138265833175
x31 = -47.5165888855456
x32 = -34.164820107789
x33 = -18.45685683984
x34 = 56.1559686829176
x35 = -62.4391539900971
x36 = -75.7909227678538
x37 = -53.7997741927252
x38 = -5.89048622548086
x39 = 100.138265833175
x40 = -41.233403578366
x41 = -109.563043793944
x42 = -68.7223392972767
x43 = -21.5984494934298
x44 = 84.4303025652257
x45 = -63.2245521534946
x46 = 31.8086256175967
x47 = 0.392699081698724
x48 = 12.1736715326604
x49 = 75.7909227678538
x50 = 69.5077374606742
x51 = 91.4988860358027
x52 = -69.5077374606742
x53 = 3.53429173528852
x54 = 16.1006623496477
x55 = -16.1006623496477
x56 = -31.8086256175967
x57 = 82.0741080750334
x58 = 44.3749962319558
x59 = -56.1559686829176
x60 = 93.8550805259951
x61 = 40.4480054149686
x62 = -87.5718952188155
x63 = -426.863901806513
x64 = -103.279858486764
x65 = 109.563043793944
x66 = 138.62277583965
x67 = 24.7400421470196
x68 = -3.53429173528852
x69 = 62.4391539900971
x70 = -9.8174770424681
x71 = -27.8816348006094
x72 = 78.1471172580461
x73 = 97.7820713429823
x74 = -12.1736715326604
x75 = -97.7820713429823
x76 = -91.4988860358027
x77 = 90.7134878724053
x78 = -90.7134878724053
x79 = -60.0829594999048
x80 = -46.7311907221482
x81 = 27.8816348006094
x82 = -2.74889357189107
x83 = 25.5254403104171
x84 = -71.8639319508665
x85 = -65.5807466436869
x86 = -25.5254403104171
x86 = -25.5254403104171
Gráfico
cos(2*x)=sqrt(2)/2 la ecuación