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-2*sin(x)+cos(x)^2=0 la ecuación

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Solución numérica:

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Solución

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

(-2)^2 - 4 * (-1) * (1) = 8

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

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

o
w1=−2−1w_{1} = - \sqrt{2} - 1
w2=−1+2w_{2} = -1 + \sqrt{2}
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⁡(−2−1)x_{1} = 2 \pi n + \operatorname{asin}{\left(- \sqrt{2} - 1 \right)}
x1=2πn−asin⁡(1+2)x_{1} = 2 \pi n - \operatorname{asin}{\left(1 + \sqrt{2} \right)}
x2=2πn+asin⁡(w2)x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}
x2=2πn+asin⁡(−1+2)x_{2} = 2 \pi n + \operatorname{asin}{\left(-1 + \sqrt{2} \right)}
x2=2πn−asin⁡(1−2)x_{2} = 2 \pi n - \operatorname{asin}{\left(1 - \sqrt{2} \right)}
x3=2πn−asin⁡(w1)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
x3=2πn+π−asin⁡(−2−1)x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(- \sqrt{2} - 1 \right)}
x3=2πn+π+asin⁡(1+2)x_{3} = 2 \pi n + \pi + \operatorname{asin}{\left(1 + \sqrt{2} \right)}
x4=2πn−asin⁡(w2)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi
x4=2πn−asin⁡(−1+2)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(-1 + \sqrt{2} \right)} + \pi
x4=2πn+asin⁡(1−2)+πx_{4} = 2 \pi n + \operatorname{asin}{\left(1 - \sqrt{2} \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*re\atan\-1 + \/ 2  + \/ 2 *\/  1 - \/ 2  // - 2*I*im\atan\-1 + \/ 2  + \/ 2 *\/  1 - \/ 2  // + 2*atan\1 + \/ 2  + \/ 2 *\/  1 + \/ 2  / + 2*re\atan\1 - \/ 2  + \/ 2 *\/  1 - \/ 2  // + 2*I*im\atan\1 - \/ 2  + \/ 2 *\/  1 - \/ 2  // + 2*atan\1 + \/ 2  - \/ 2 *\/  1 + \/ 2  /
((2atan⁡(1+2+21+2)+(−2re⁡(atan⁡(−1+2+21−2))−2iim⁡(atan⁡(−1+2+21−2))))+(2re⁡(atan⁡(−2+1+21−2))+2iim⁡(atan⁡(−2+1+21−2))))+2atan⁡(−21+2+1+2)\left(\left(2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{1 + \sqrt{2}} \right)} + \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(-1 + \sqrt{2} + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(-1 + \sqrt{2} + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)}\right)\right) + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)}\right)\right) + 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{1 + \sqrt{2}} + 1 + \sqrt{2} \right)} 
=
      /    /                      ___________\\         /                     ___________\         /                     ___________\       /    /                     ___________\\         /    /                      ___________\\         /    /                     ___________\\
      |    |       ___     ___   /       ___ ||         |      ___     ___   /       ___ |         |      ___     ___   /       ___ |       |    |      ___     ___   /       ___ ||         |    |       ___     ___   /       ___ ||         |    |      ___     ___   /       ___ ||
- 2*re\atan\-1 + \/ 2  + \/ 2 *\/  1 - \/ 2  // + 2*atan\1 + \/ 2  + \/ 2 *\/  1 + \/ 2  / + 2*atan\1 + \/ 2  - \/ 2 *\/  1 + \/ 2  / + 2*re\atan\1 - \/ 2  + \/ 2 *\/  1 - \/ 2  // - 2*I*im\atan\-1 + \/ 2  + \/ 2 *\/  1 - \/ 2  // + 2*I*im\atan\1 - \/ 2  + \/ 2 *\/  1 - \/ 2  //
−2re⁡(atan⁡(−1+2+21−2))+2re⁡(atan⁡(−2+1+21−2))+2atan⁡(−21+2+1+2)+2atan⁡(1+2+21+2)−2iim⁡(atan⁡(−1+2+21−2))+2iim⁡(atan⁡(−2+1+21−2))- 2 \operatorname{re}{\left(\operatorname{atan}{\left(-1 + \sqrt{2} + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} + 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{1 + \sqrt{2}} + 1 + \sqrt{2} \right)} + 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{1 + \sqrt{2}} \right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(-1 + \sqrt{2} + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} 
producto
/      /    /                      ___________\\         /    /                      ___________\\\       /                     ___________\ /    /    /                     ___________\\         /    /                     ___________\\\       /                     ___________\
|      |    |       ___     ___   /       ___ ||         |    |       ___     ___   /       ___ |||       |      ___     ___   /       ___ | |    |    |      ___     ___   /       ___ ||         |    |      ___     ___   /       ___ |||       |      ___     ___   /       ___ |
\- 2*re\atan\-1 + \/ 2  + \/ 2 *\/  1 - \/ 2  // - 2*I*im\atan\-1 + \/ 2  + \/ 2 *\/  1 - \/ 2  ///*2*atan\1 + \/ 2  + \/ 2 *\/  1 + \/ 2  /*\2*re\atan\1 - \/ 2  + \/ 2 *\/  1 - \/ 2  // + 2*I*im\atan\1 - \/ 2  + \/ 2 *\/  1 - \/ 2  ///*2*atan\1 + \/ 2  - \/ 2 *\/  1 + \/ 2  /
(−2re⁡(atan⁡(−1+2+21−2))−2iim⁡(atan⁡(−1+2+21−2)))2atan⁡(1+2+21+2)(2re⁡(atan⁡(−2+1+21−2))+2iim⁡(atan⁡(−2+1+21−2)))2atan⁡(−21+2+1+2)\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(-1 + \sqrt{2} + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(-1 + \sqrt{2} + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)}\right) 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{1 + \sqrt{2}} \right)} \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)}\right) 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{1 + \sqrt{2}} + 1 + \sqrt{2} \right)} 
=
    /    /    /                     ___________\\     /    /                     ___________\\\ /    /    /                      ___________\\     /    /                      ___________\\\     /                     ___________\     /                     ___________\
    |    |    |      ___     ___   /       ___ ||     |    |      ___     ___   /       ___ ||| |    |    |       ___     ___   /       ___ ||     |    |       ___     ___   /       ___ |||     |      ___     ___   /       ___ |     |      ___     ___   /       ___ |
-16*\I*im\atan\1 - \/ 2  + \/ 2 *\/  1 - \/ 2  // + re\atan\1 - \/ 2  + \/ 2 *\/  1 - \/ 2  ///*\I*im\atan\-1 + \/ 2  + \/ 2 *\/  1 - \/ 2  // + re\atan\-1 + \/ 2  + \/ 2 *\/  1 - \/ 2  ///*atan\1 + \/ 2  + \/ 2 *\/  1 + \/ 2  /*atan\1 + \/ 2  - \/ 2 *\/  1 + \/ 2  /
−16(re⁡(atan⁡(−1+2+21−2))+iim⁡(atan⁡(−1+2+21−2)))(re⁡(atan⁡(−2+1+21−2))+iim⁡(atan⁡(−2+1+21−2)))atan⁡(1+2+21+2)atan⁡(−21+2+1+2)- 16 \left(\operatorname{re}{\left(\operatorname{atan}{\left(-1 + \sqrt{2} + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(-1 + \sqrt{2} + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)}\right) \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{1 + \sqrt{2}} \right)} \operatorname{atan}{\left(- \sqrt{2} \sqrt{1 + \sqrt{2}} + 1 + \sqrt{2} \right)} 
-16*(i*im(atan(1 - sqrt(2) + sqrt(2)*sqrt(1 - sqrt(2)))) + re(atan(1 - sqrt(2) + sqrt(2)*sqrt(1 - sqrt(2)))))*(i*im(atan(-1 + sqrt(2) + sqrt(2)*sqrt(1 - sqrt(2)))) + re(atan(-1 + sqrt(2) + sqrt(2)*sqrt(1 - sqrt(2)))))*atan(1 + sqrt(2) + sqrt(2)*sqrt(1 + sqrt(2)))*atan(1 + sqrt(2) - sqrt(2)*sqrt(1 + sqrt(2)))
Respuesta rápida [src] 
           /    /                      ___________\\         /    /                      ___________\\
           |    |       ___     ___   /       ___ ||         |    |       ___     ___   /       ___ ||
x1 = - 2*re\atan\-1 + \/ 2  + \/ 2 *\/  1 - \/ 2  // - 2*I*im\atan\-1 + \/ 2  + \/ 2 *\/  1 - \/ 2  //
x1=−2re⁡(atan⁡(−1+2+21−2))−2iim⁡(atan⁡(−1+2+21−2))x_{1} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(-1 + \sqrt{2} + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(-1 + \sqrt{2} + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} 
           /                     ___________\
           |      ___     ___   /       ___ |
x2 = 2*atan\1 + \/ 2  + \/ 2 *\/  1 + \/ 2  /
x2=2atan⁡(1+2+21+2)x_{2} = 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{1 + \sqrt{2}} \right)} 
         /    /                     ___________\\         /    /                     ___________\\
         |    |      ___     ___   /       ___ ||         |    |      ___     ___   /       ___ ||
x3 = 2*re\atan\1 - \/ 2  + \/ 2 *\/  1 - \/ 2  // + 2*I*im\atan\1 - \/ 2  + \/ 2 *\/  1 - \/ 2  //
x3=2re⁡(atan⁡(−2+1+21−2))+2iim⁡(atan⁡(−2+1+21−2))x_{3} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} 
           /                     ___________\
           |      ___     ___   /       ___ |
x4 = 2*atan\1 + \/ 2  - \/ 2 *\/  1 + \/ 2  /
x4=2atan⁡(−21+2+1+2)x_{4} = 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{1 + \sqrt{2}} + 1 + \sqrt{2} \right)} 
x4 = 2*atan(-sqrt(2)*sqrt(1 + sqrt(2)) + 1 + sqrt(2))
Respuesta numérica [src] 
x1 = -16.1350418543414
x2 = 19.2766345079312
x3 = 6.71026389357206
x4 = 94.6748581940863
x5 = -22.418227161521
x6 = 21.5640699887361
x7 = 100.958043501266
x8 = 63.2589316581883
x9 = 90.6791083677115
x10 = 84.3959230605319
x11 = -91.5332655404965
x12 = 12.9934492007516
x13 = 44.4093757366496
x14 = 34.1304406030953
x15 = -53.834153697419
x16 = 71.8295524461728
x17 = -30.9888479495055
x18 = -28.7014124687006
x19 = 59.2631818318136
x20 = 0.427078586392476
x21 = -34.9845977758802
x22 = 78.1127377533523
x23 = 46.6968112174544
x24 = -122.949192076394
x25 = -72.6837096189577
x26 = 96.9622936748911
x27 = -56.1215891782238
x28 = -47.5509683902394
x29 = -85.2500802333169
x30 = 38.12619042947
x31 = 27.8472552959157
x32 = -631.887201957941
x33 = 15.2808846815565
x34 = -68.687959792583
x35 = -49.8384038710442
x36 = -37.272033256685
x37 = -12.1392920279667
x38 = 75.8253022725475
x39 = -74.9711450997626
x40 = 8.9976993743769
x41 = -87.5375157141217
x42 = -100.103886328481
x43 = 2.71451406719732
x44 = -110.382821462035
x45 = 65.5463671389932
x46 = -18.4224773351463
x47 = 82.1084875797271
x48 = -43.5552185638646
x49 = -9.85185654716186
x50 = 88.3916728869067
x51 = 56.9757463510088
x52 = 40.4136259102748
x53 = -66.4005243117781
x54 = -78.9668949261373
x55 = -81.2543304069421
x56 = -97.8164508476761
x57 = -93.8207010213013
x58 = -41.2677830830598
x59 = -3.56867123998227
x60 = 69.5421169653679
x61 = 31.8430051222904
x62 = 50.6925610438292
x63 = -60.1173390045985
x64 = -5.85610672078711
x65 = -62.4047744854034
x66 = 25.5598198151108
x67 = 52.979996524634
x68 = -24.7056626423259
x68 = -24.7056626423259
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