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sin(x)^(2)-2*sin(x)-3=0

sin(x)^(2)-2*sin(x)-3=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   2                      
sin (x) - 2*sin(x) - 3 = 0
(sin⁡2(x)−2sin⁡(x))−3=0\left(\sin^{2}{\left(x \right)} - 2 \sin{\left(x \right)}\right) - 3 = 0
Simplificar
Solución detallada
Tenemos la ecuación
(sin⁡2(x)−2sin⁡(x))−3=0\left(\sin^{2}{\left(x \right)} - 2 \sin{\left(x \right)}\right) - 3 = 0
cambiamos
sin⁡2(x)−2sin⁡(x)−3=0\sin^{2}{\left(x \right)} - 2 \sin{\left(x \right)} - 3 = 0
(sin⁡2(x)−2sin⁡(x))−3=0\left(\sin^{2}{\left(x \right)} - 2 \sin{\left(x \right)}\right) - 3 = 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=−3c = -3
, entonces
D = b^2 - 4 * a * c = 

(-2)^2 - 4 * (1) * (-3) = 16

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

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

o
w1=3w_{1} = 3
w2=−1w_{2} = -1
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⁡(3)x_{1} = 2 \pi n + \operatorname{asin}{\left(3 \right)}
x1=2πn+asin⁡(3)x_{1} = 2 \pi n + \operatorname{asin}{\left(3 \right)}
x2=2πn+asin⁡(w2)x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}
x2=2πn+asin⁡(−1)x_{2} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}
x2=2πn−π2x_{2} = 2 \pi n - \frac{\pi}{2}
x3=2πn−asin⁡(w1)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
x3=2πn+π−asin⁡(3)x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}
x3=2πn+π−asin⁡(3)x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}
x4=2πn−asin⁡(w2)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi
x4=2πn−asin⁡(−1)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi
x4=2πn+3π2x_{4} = 2 \pi n + \frac{3 \pi}{2}
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
  pi   3*pi                                                                 
- -- + ---- + pi - re(asin(3)) - I*im(asin(3)) + I*im(asin(3)) + re(asin(3))
  2     2
(re⁡(asin⁡(3))+iim⁡(asin⁡(3)))+((−π2+3π2)+(−re⁡(asin⁡(3))+π−iim⁡(asin⁡(3))))\left(\operatorname{re}{\left(\operatorname{asin}{\left(3 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(3 \right)}\right)}\right) + \left(\left(- \frac{\pi}{2} + \frac{3 \pi}{2}\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(3 \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(3 \right)}\right)}\right)\right) 
=
2*pi
2π2 \pi 
producto
-pi  3*pi                                                                 
----*----*(pi - re(asin(3)) - I*im(asin(3)))*(I*im(asin(3)) + re(asin(3)))
 2    2
−π23π2(−re⁡(asin⁡(3))+π−iim⁡(asin⁡(3)))(re⁡(asin⁡(3))+iim⁡(asin⁡(3)))- \frac{\pi}{2} \frac{3 \pi}{2} \left(- \operatorname{re}{\left(\operatorname{asin}{\left(3 \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(3 \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(3 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(3 \right)}\right)}\right) 
=
    2                                                                  
3*pi *(I*im(asin(3)) + re(asin(3)))*(-pi + I*im(asin(3)) + re(asin(3)))
-----------------------------------------------------------------------
                                   4
3π2(re⁡(asin⁡(3))+iim⁡(asin⁡(3)))(−π+re⁡(asin⁡(3))+iim⁡(asin⁡(3)))4\frac{3 \pi^{2} \left(\operatorname{re}{\left(\operatorname{asin}{\left(3 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(3 \right)}\right)}\right) \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(3 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(3 \right)}\right)}\right)}{4} 
3*pi^2*(i*im(asin(3)) + re(asin(3)))*(-pi + i*im(asin(3)) + re(asin(3)))/4
Respuesta rápida [src] 
     -pi 
x1 = ----
      2
x1=−π2x_{1} = - \frac{\pi}{2} 
     3*pi
x2 = ----
      2
x2=3π2x_{2} = \frac{3 \pi}{2} 
x3 = pi - re(asin(3)) - I*im(asin(3))
x3=−re⁡(asin⁡(3))+π−iim⁡(asin⁡(3))x_{3} = - \operatorname{re}{\left(\operatorname{asin}{\left(3 \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(3 \right)}\right)} 
x4 = I*im(asin(3)) + re(asin(3))
x4=re⁡(asin⁡(3))+iim⁡(asin⁡(3))x_{4} = \operatorname{re}{\left(\operatorname{asin}{\left(3 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(3 \right)}\right)} 
x4 = re(asin(3)) + i*im(asin(3))
Respuesta numérica [src] 
x1 = -51.8362786895378
x2 = 48.6946859120413
x3 = 17.2787599090696
x4 = 80.1106131400977
x5 = -1.57079643080582
x6 = 73.827427591263
x7 = 4.71238923181769
x8 = 29.8451303217623
x9 = -39.2699084013107
x10 = -58.119463652249
x11 = 48.6946863700612
x12 = -20.4203520186622
x13 = -70.6858350312864
x14 = -14.1371668194089
x15 = 86.3937976360352
x16 = -64.4026496039382
x17 = 92.6769835070253
x18 = 882955.610865625
x19 = 4.71238875528975
x20 = -83.2522055577573
x21 = -83.2522051037983
x22 = -64.4026491754768
x23 = 98.9601682894894
x24 = -14.1371665766936
x25 = 61.261057062553
x26 = 117.809725233756
x27 = 212.057503588662
x28 = 67.5442422944741
x29 = -20.4203524688757
x30 = 23.5619446676752
x31 = 42.4115007283113
x32 = 42.4115009058131
x33 = -83.2522050480464
x34 = 92.6769830689412
x35 = 54.9778717720118
x36 = 42.4115005526819
x37 = -133.517687838813
x38 = 10.9955746230205
x39 = -26.7035372446302
x40 = 73.8274274813446
x41 = 54.9778711353875
x42 = -1.57079653638307
x43 = -39.2699079075865
x44 = 67.5442418055755
x45 = 17.2787592660486
x46 = -39.2699076412407
x47 = -108.38494727522
x48 = -139.800873467486
x49 = -32.9867225263379
x50 = 61.2610564153358
x51 = -14.1371668381663
x52 = -76.9690203256113
x53 = 10.9955739814993
x54 = 98.9601689206173
x55 = 36.1283157033748
x56 = -89.5353907308516
x57 = -26.703537882833
x58 = -76.9690196760468
x59 = 86.3937978876249
x60 = -45.5530936309972
x61 = -164.933614398177
x62 = -58.1194639985047
x63 = -32.9867231721691
x64 = 4.71238886235498
x65 = -95.8185758680893
x66 = -45.5530935897427
x67 = -89.5353907485116
x68 = -7.85398149759801
x69 = 23.5619451379884
x70 = -70.6858343985417
x71 = 29.845130504401
x71 = 29.845130504401
Gráfico
sin(x)^(2)-2*sin(x)-3=0 la ecuación
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