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sin(x)^2=1

sin(x)^2=1 la ecuación

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Solución numérica:

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Solución

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

(0)^2 - 4 * (1) * (-1) = 4

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

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

o
w1=1w_{1} = 1
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⁡(1)x_{1} = 2 \pi n + \operatorname{asin}{\left(1 \right)}
x1=2πn+π2x_{1} = 2 \pi n + \frac{\pi}{2}
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−asin⁡(1)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi
x3=2πn+π2x_{3} = 2 \pi n + \frac{\pi}{2}
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
0-80-60-40-2020406080-10010002
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
     -pi 
x1 = ----
      2
x1=−π2x_{1} = - \frac{\pi}{2} 
     pi
x2 = --
     2
x2=π2x_{2} = \frac{\pi}{2} 
     3*pi
x3 = ----
      2
x3=3π2x_{3} = \frac{3 \pi}{2} 
x3 = 3*pi/2
Suma y producto de raíces [src] 
suma
  pi   pi   3*pi
- -- + -- + ----
  2    2     2
(−π2+π2)+3π2\left(- \frac{\pi}{2} + \frac{\pi}{2}\right) + \frac{3 \pi}{2} 
=
3*pi
----
 2
3π2\frac{3 \pi}{2} 
producto
-pi  pi 3*pi
----*--*----
 2   2   2
3π2⋅−π2π2\frac{3 \pi}{2} \cdot - \frac{\pi}{2} \frac{\pi}{2} 
=
     3
-3*pi 
------
  8
−3π38- \frac{3 \pi^{3}}{8} 
-3*pi^3/8
Respuesta numérica [src] 
x1 = 20.4203521497111
x2 = 42.4115007291722
x3 = -58.1194639993376
x4 = -7.85398149857354
x5 = -95.8185758681287
x6 = -17.2787590276524
x7 = 39.2699081179815
x8 = 76.9690197631883
x9 = 36.1283156002139
x10 = -70.6858346386357
x11 = -86.393797765473
x12 = -98.96016883042
x13 = -20.4203520321877
x14 = 92.6769830795146
x15 = 64.4026493086922
x16 = -45.5530935883361
x17 = -98.960168684456
x18 = -32.9867231091652
x19 = 61.2610569989704
x20 = -4.71238872430683
x21 = 83.2522052340866
x22 = 95.8185760590309
x23 = -83.2522055415057
x24 = -54.9778713137198
x25 = 32.986722928111
x26 = 14.1371671048484
x27 = -39.2699083866483
x28 = -80.1106125795659
x29 = -54.9778716831146
x30 = 7.85398174058521
x31 = -10.9955741902138
x32 = -92.6769830239371
x33 = -26.7035375427973
x34 = -36.1283154192437
x35 = 98.9601685932308
x36 = -76.9690202568697
x37 = 89.5353908552844
x38 = 86.393797888273
x39 = -92.6769831823972
x40 = 80.1106126771746
x41 = -14.1371668392726
x42 = -67.5442421675773
x43 = 76.9690207492347
x44 = -76.9690198771149
x45 = 67.5442422779275
x46 = 23.5619449395428
x47 = -39.2699081528781
x48 = 83.2522055730903
x49 = 39.2699084246933
x50 = -1.57079642969308
x51 = 29.845130320338
x52 = -32.9867227513827
x53 = 32.9867226137576
x54 = -23.5619450090417
x55 = -26.7035372990183
x56 = 54.9778711883962
x57 = 1.5707965454425
x58 = -42.4115006098842
x59 = -29.8451300963672
x60 = -48.6946860920117
x61 = -51.8362786897497
x62 = -48.6946858738636
x63 = 48.6946859238715
x64 = 51.8362788999928
x65 = -17.2787598091171
x66 = 61.2610566752601
x67 = 17.2787595624179
x68 = 45.553093700501
x69 = 26.7035373461441
x70 = 23.5619451230057
x71 = 10.9955743696636
x72 = -61.2610569641117
x73 = 98.9601683381274
x74 = -70.685834448838
x75 = 10.9955740392793
x76 = -64.4026491876462
x77 = 70.6858345016621
x78 = -61.2610562242523
x79 = 80.1106131434937
x80 = -89.5353907467661
x81 = 17.2787598502655
x82 = 541.924732890135
x83 = -73.8274272800405
x84 = 54.9778714849733
x85 = 76.9690200400775
x86 = 4.71238876848081
x87 = 58.1194644379895
x88 = 73.8274274795554
x89 = -4.7123889912442
x90 = -10.9955745350309
x91 = -98.9601684414698
x91 = -98.9601684414698
Gráfico
sin(x)^2=1 la ecuación
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