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sin^2(x)+1/2(sin2(x))-2cos^2(x)=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
             2                   
   2      sin (x)        2       
sin (x) + ------- - 2*cos (x) = 0
             2
(sin⁡2(x)2+sin⁡2(x))−2cos⁡2(x)=0\left(\frac{\sin^{2}{\left(x \right)}}{2} + \sin^{2}{\left(x \right)}\right) - 2 \cos^{2}{\left(x \right)} = 0
Simplificar
Solución detallada
Tenemos la ecuación
(sin⁡2(x)2+sin⁡2(x))−2cos⁡2(x)=0\left(\frac{\sin^{2}{\left(x \right)}}{2} + \sin^{2}{\left(x \right)}\right) - 2 \cos^{2}{\left(x \right)} = 0
cambiamos
7sin⁡2(x)2−2=0\frac{7 \sin^{2}{\left(x \right)}}{2} - 2 = 0
7sin⁡2(x)2−2=0\frac{7 \sin^{2}{\left(x \right)}}{2} - 2 = 0
Sustituimos
w=sin⁡(x)w = \sin{\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=72a = \frac{7}{2}
b=0b = 0
c=−2c = -2
, entonces
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (7/2) * (-2) = 28

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

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

o
w1=277w_{1} = \frac{2 \sqrt{7}}{7}
w2=−277w_{2} = - \frac{2 \sqrt{7}}{7}
hacemos cambio inverso
sin⁡(x)=w\sin{\left(x \right)} = w
Tenemos la ecuación
sin⁡(x)=w\sin{\left(x \right)} = w
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
x=2πn+asin⁡(w)x = 2 \pi n + \operatorname{asin}{\left(w \right)}
x=2πn−asin⁡(w)+πx = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
O
x=2πn+asin⁡(w)x = 2 \pi n + \operatorname{asin}{\left(w \right)}
x=2πn−asin⁡(w)+πx = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
, donde n es cualquier número entero
sustituimos w:
x1=2πn+asin⁡(w1)x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}
x1=2πn+asin⁡(277)x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{2 \sqrt{7}}{7} \right)}
x1=2πn+asin⁡(277)x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{2 \sqrt{7}}{7} \right)}
x2=2πn+asin⁡(w2)x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}
x2=2πn+asin⁡(−277)x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{2 \sqrt{7}}{7} \right)}
x2=2πn−asin⁡(277)x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{2 \sqrt{7}}{7} \right)}
x3=2πn−asin⁡(w1)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
x3=2πn−asin⁡(277)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{2 \sqrt{7}}{7} \right)} + \pi
x3=2πn−asin⁡(277)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{2 \sqrt{7}}{7} \right)} + \pi
x4=2πn−asin⁡(w2)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi
x4=2πn−asin⁡(−277)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{2 \sqrt{7}}{7} \right)} + \pi
x4=2πn+asin⁡(277)+πx_{4} = 2 \pi n + \operatorname{asin}{\left(\frac{2 \sqrt{7}}{7} \right)} + \pi
Gráfica
0-80-60-40-2020406080-1001005-5
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
      /    ___\       /    ___\
      |2*\/ 3 |       |2*\/ 3 |
- atan|-------| + atan|-------|
      \   3   /       \   3   /
−atan⁡(233)+atan⁡(233)- \operatorname{atan}{\left(\frac{2 \sqrt{3}}{3} \right)} + \operatorname{atan}{\left(\frac{2 \sqrt{3}}{3} \right)} 
=
0
00 
producto
     /    ___\     /    ___\
     |2*\/ 3 |     |2*\/ 3 |
-atan|-------|*atan|-------|
     \   3   /     \   3   /
−atan⁡(233)atan⁡(233)- \operatorname{atan}{\left(\frac{2 \sqrt{3}}{3} \right)} \operatorname{atan}{\left(\frac{2 \sqrt{3}}{3} \right)} 
=
      /    ___\
     2|2*\/ 3 |
-atan |-------|
      \   3   /
−atan⁡2(233)- \operatorname{atan}^{2}{\left(\frac{2 \sqrt{3}}{3} \right)} 
-atan(2*sqrt(3)/3)^2
Respuesta rápida [src] 
          /    ___\
          |2*\/ 3 |
x1 = -atan|-------|
          \   3   /
x1=−atan⁡(233)x_{1} = - \operatorname{atan}{\left(\frac{2 \sqrt{3}}{3} \right)} 
         /    ___\
         |2*\/ 3 |
x2 = atan|-------|
         \   3   /
x2=atan⁡(233)x_{2} = \operatorname{atan}{\left(\frac{2 \sqrt{3}}{3} \right)} 
x2 = atan(2*sqrt(3)/3)
Respuesta numérica [src] 
x1 = -17.9924839736886
x2 = -11.709298666509
x3 = -19.7066278693889
x4 = -77.6827443918947
x5 = -61.9747811239457
x6 = -25.9898131765685
x7 = 60.5473323660562
x8 = -2.28452070573966
x9 = 68.2579664311253
x10 = 46.2668178559968
x11 = -91.9632589019541
x12 = -46.2668178559968
x13 = -69.9721103268256
x14 = -27.417261934458
x15 = 54.2641470588766
x16 = -13.4234425622093
x17 = 33.7004472416376
x18 = -60.5473323660562
x19 = -71.3995590847151
x20 = -57.4057397124664
x21 = 3.99866460143992
x22 = 82.5384809411848
x23 = -10.2818499086195
x24 = -54.2641470588766
x25 = 61.9747811239457
x26 = -24.2756692808682
x27 = 77.6827443918947
x28 = -99.6738929670232
x29 = 10.2818499086195
x30 = -76.2552956340052
x31 = 47.980961751697
x32 = -55.6915958167661
x33 = -5.42611335932946
x34 = 8.56770601291925
x35 = -85.6800735947746
x36 = -16.5650352157991
x37 = 98.2464442091337
x38 = -79.396888287595
x39 = 30.5588545880478
x40 = 11.709298666509
x41 = 63.688925019646
x42 = -36.8420398952274
x43 = 19.7066278693889
x44 = -82.5384809411848
x45 = -49.4084105095866
x46 = -90.2491150062539
x47 = -93.3907076598437
x48 = 25.9898131765685
x49 = 96.5323003134335
x50 = -38.5561837909276
x51 = 139.087148705801
x52 = 55.6915958167661
x53 = -39.9836325488172
x54 = 247.328747685744
x55 = 90.2491150062539
x56 = 10595.7349486105
x57 = 16.5650352157991
x58 = 32.2729984837481
x59 = 39.9836325488172
x60 = 38.5561837909276
x61 = 91.9632589019541
x62 = -63.688925019646
x63 = -47.980961751697
x64 = 85.6800735947746
x65 = -68.2579664311253
x66 = 76.2552956340052
x67 = -41.6977764445174
x68 = 41.6977764445174
x69 = -35.4145911373379
x70 = 24.2756692808682
x71 = 2.28452070573966
x72 = -32.2729984837481
x73 = 69.9721103268256
x74 = 83.9659296990743
x75 = -83.9659296990743
x76 = 74.5411517383049
x77 = -98.2464442091337
x78 = -3.99866460143992
x79 = 99.6738929670232
x80 = 17.9924839736886
x81 = -374.706597725036
x82 = -33.7004472416376
x83 = 52.5500031631764
x83 = 52.5500031631764
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