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2sin^2(x)-cos(x)=1

2sin^2(x)-cos(x)=1 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
     2                
2*sin (x) - cos(x) = 1
$$2 \sin^{2}{\left(x \right)} - \cos{\left(x \right)} = 1$$
Solución detallada
Tenemos la ecuación
$$2 \sin^{2}{\left(x \right)} - \cos{\left(x \right)} = 1$$
cambiamos
$$- \cos{\left(x \right)} - \cos{\left(2 x \right)} = 0$$
$$- 2 \cos^{2}{\left(x \right)} - \cos{\left(x \right)} + 1 = 0$$
Sustituimos
$$w = \cos{\left(x \right)}$$
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:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = -2$$
$$b = -1$$
$$c = 1$$
, entonces
D = b^2 - 4 * a * c = 

(-1)^2 - 4 * (-2) * (1) = 9

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

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

o
$$w_{1} = -1$$
$$w_{2} = \frac{1}{2}$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(-1 \right)}$$
$$x_{1} = \pi n + \pi$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$x_{2} = \pi n + \frac{\pi}{3}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(-1 \right)}$$
$$x_{3} = \pi n$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$x_{4} = \pi n - \frac{2 \pi}{3}$$
Gráfica
Suma y producto de raíces [src]
suma
  pi   pi
- -- + --
  3    3 
$$- \frac{\pi}{3} + \frac{\pi}{3}$$
=
0
$$0$$
producto
-pi  pi
----*--
 3   3 
$$- \frac{\pi}{3} \frac{\pi}{3}$$
=
   2 
-pi  
-----
  9  
$$- \frac{\pi^{2}}{9}$$
-pi^2/9
Respuesta rápida [src]
     -pi 
x1 = ----
      3  
$$x_{1} = - \frac{\pi}{3}$$
     pi
x2 = --
     3 
$$x_{2} = \frac{\pi}{3}$$
x2 = pi/3
Respuesta numérica [src]
x1 = 26.1799387799149
x2 = 11.5191730631626
x3 = 91.1061868116125
x4 = 3.14159271706432
x5 = -51.3126800086333
x6 = 3.14159267447126
x7 = 107.86134777325
x8 = -26.1799387799149
x9 = 99.4837673636768
x10 = -76.4454212373516
x11 = -13.6135681655558
x12 = -7.33038285837618
x13 = 47.1238897752019
x14 = 65.9734457528689
x15 = 13.6135681655558
x16 = 68.0678408277789
x17 = 40.8407042778045
x18 = -5.23598775598299
x19 = 21.9911485851931
x20 = 78.5398161904624
x21 = 28.2743338652086
x22 = -99.4837673636768
x23 = -84.8230014829768
x24 = 34.5575190335478
x25 = 36.6519142918809
x26 = -93.2005820564972
x27 = 91.1061869261407
x28 = 47.1238894268221
x29 = 15.7079634367135
x30 = 55.5014702134197
x31 = -28.2743337200245
x32 = 91.1061863890352
x33 = 97.389372486408
x34 = -19.8967534727354
x35 = -40.8407049942712
x36 = -9.42477812311019
x37 = 59.6902605931502
x38 = -55.5014702134197
x39 = 17.8023583703422
x40 = 61.7846555205993
x41 = 3873.5837416047
x42 = -68.0678408277789
x43 = 21.9911485973609
x44 = 24.0855436775217
x45 = 30.3687289847013
x46 = 70.162235930172
x47 = 9.42477818680547
x48 = -11.5191730631626
x49 = -40.8407047547408
x50 = 82.7286065445312
x51 = 47.1238898268985
x52 = -84.8230022421807
x53 = -15.7079632965016
x54 = 3.14159322994749
x55 = 63.8790506229925
x56 = 47.1238901206303
x57 = 76.4454212373516
x58 = -17.8023583703422
x59 = -65.9734457650482
x60 = 57.5958653158129
x61 = -84.8230015251551
x62 = -91.1061871711313
x63 = -53.4070752795041
x64 = -57.5958653158129
x65 = -74.3510261349584
x66 = -21.9911485864549
x67 = 53.4070753369186
x68 = 84.8230014287926
x69 = -32.4631240870945
x70 = -40.8407044128941
x71 = -719.424718069224
x72 = -95.2949771588904
x73 = -47.1238905036874
x74 = 91.1061868861836
x75 = 19.8967534727354
x76 = 32.4631240870945
x77 = -30.3687289847013
x78 = -49.2182849062401
x79 = -97.3893724356252
x80 = -78.5398161151012
x81 = -34.5575189638817
x82 = -40.8407044009017
x83 = 80.634211442138
x84 = -47.1238900222279
x85 = 72.2566310277195
x86 = -24.0855436775217
x87 = 3.14159276530697
x88 = -70.162235930172
x89 = -3.14159287255706
x90 = -59.690260457585
x91 = 38.7463093942741
x92 = 74.3510261349584
x93 = -61.7846555205993
x94 = -63.8790506229925
x95 = -72.256630877064
x95 = -72.256630877064
Gráfico
2sin^2(x)-cos(x)=1 la ecuación