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sin(2x)=sqrt(2)/2

sin(2x)=sqrt(2)/2 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
             ___
           \/ 2 
sin(2*x) = -----
             2  
$$\sin{\left(2 x \right)} = \frac{\sqrt{2}}{2}$$
Solución detallada
Tenemos la ecuación
$$\sin{\left(2 x \right)} = \frac{\sqrt{2}}{2}$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$2 x = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$2 x = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)} + \pi$$
O
$$2 x = 2 \pi n + \frac{\pi}{4}$$
$$2 x = 2 \pi n + \frac{3 \pi}{4}$$
, donde n es cualquier número entero
Dividamos ambos miembros de la ecuación obtenida en
$$2$$
obtenemos la respuesta:
$$x_{1} = \pi n + \frac{\pi}{8}$$
$$x_{2} = \pi n + \frac{3 \pi}{8}$$
Gráfica
Respuesta rápida [src]
     pi
x1 = --
     8 
$$x_{1} = \frac{\pi}{8}$$
     3*pi
x2 = ----
      8  
$$x_{2} = \frac{3 \pi}{8}$$
x2 = 3*pi/8
Suma y producto de raíces [src]
suma
pi   3*pi
-- + ----
8     8  
$$\frac{\pi}{8} + \frac{3 \pi}{8}$$
=
pi
--
2 
$$\frac{\pi}{2}$$
producto
pi 3*pi
--*----
8   8  
$$\frac{\pi}{8} \frac{3 \pi}{8}$$
=
    2
3*pi 
-----
  64 
$$\frac{3 \pi^{2}}{64}$$
3*pi^2/64
Respuesta numérica [src]
x1 = -31.0232274541992
x2 = 100.923663996572
x3 = 70.2931356240716
x4 = 88.3572933822129
x5 = -61.6537558266997
x6 = 66.3661448070844
x7 = -9.03207887907065
x8 = -58.5121631731099
x9 = 53.7997741927252
x10 = 56.941366846315
x11 = 4.31968989868597
x12 = -75.0055246044563
x13 = -93.8550805259951
x14 = 73.4347282776614
x15 = -78.1471172580461
x16 = -15.3152641862502
x17 = 78.9325154214436
x18 = 60.0829594999048
x19 = 22.3838476568273
x20 = 94.6404786893925
x21 = -80.5033117482384
x22 = 42.0188017417635
x23 = 29.4524311274043
x24 = 9.8174770424681
x25 = -49.872783375738
x26 = -59.2975613365073
x27 = -96.2112750161874
x28 = 79.717913584841
x29 = 6.67588438887831
x30 = 95.42587685279
x31 = -74.2201264410589
x32 = 4622.46089067568
x33 = 12.9590696960579
x34 = 38.0918109247762
x35 = 51.4435797025329
x36 = 104.850654813559
x37 = -23.9546439836222
x38 = -43.5895980685584
x39 = -81.2887099116359
x40 = 34.9502182711865
x41 = -100.138265833175
x42 = 26.3108384738145
x43 = -34.164820107789
x44 = -5.89048622548086
x45 = -99.3528676697772
x46 = -21.5984494934298
x47 = 31.8086256175967
x48 = -55.3705705195201
x49 = 7.46128255227576
x50 = -53.0143760293278
x51 = 0.392699081698724
x52 = 20.0276531666349
x53 = -37.3064127613788
x54 = 75.7909227678538
x55 = 27323.6094055155
x56 = 72.649330114264
x57 = -1.96349540849362
x58 = 16.1006623496477
x59 = -67.9369411338793
x60 = 48.3019870489431
x61 = 86.0010988920206
x62 = -83.6449044018282
x63 = 82.0741080750334
x64 = 44.3749962319558
x65 = -39.6626072515711
x66 = -56.1559686829176
x67 = 89.1426915456104
x68 = -87.5718952188155
x69 = 57.7267650097125
x70 = 28.6670329640069
x71 = -8.24668071567321
x72 = -27.8816348006094
x73 = 50.6581815391354
x74 = -36.5210145979813
x75 = 13.7444678594553
x76 = -17.6714586764426
x77 = 97.7820713429823
x78 = -12.1736715326604
x79 = -14.5298660228528
x80 = -5901.87449885013
x81 = 35.7356164345839
x82 = -30.2378292908018
x83 = -45.9457925587507
x84 = 92.2842841992002
x85 = -71.8639319508665
x86 = -89.9280897090078
x87 = -52.2289778659303
x88 = -65.5807466436869
x89 = 64.009950316892
x89 = 64.009950316892
Gráfico
sin(2x)=sqrt(2)/2 la ecuación