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1-sin(a)^(2)+cos(a)^(2)=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
       2         2       
1 - sin (a) + cos (a) = 0
(1−sin⁡2(a))+cos⁡2(a)=0\left(1 - \sin^{2}{\left(a \right)}\right) + \cos^{2}{\left(a \right)} = 0
Simplificar
Solución detallada
Tenemos la ecuación
(1−sin⁡2(a))+cos⁡2(a)=0\left(1 - \sin^{2}{\left(a \right)}\right) + \cos^{2}{\left(a \right)} = 0
cambiamos
2cos⁡2(a)=02 \cos^{2}{\left(a \right)} = 0
2−2sin⁡2(a)=02 - 2 \sin^{2}{\left(a \right)} = 0
Sustituimos
w=sin⁡(a)w = \sin{\left(a \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=−2a = -2
b=0b = 0
c=2c = 2
, entonces
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (-2) * (2) = 16

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⁡(a)=w\sin{\left(a \right)} = w
Tenemos la ecuación
sin⁡(a)=w\sin{\left(a \right)} = w
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
a=2πn+asin⁡(w)a = 2 \pi n + \operatorname{asin}{\left(w \right)}
a=2πn−asin⁡(w)+πa = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
O
a=2πn+asin⁡(w)a = 2 \pi n + \operatorname{asin}{\left(w \right)}
a=2πn−asin⁡(w)+πa = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
, donde n es cualquier número entero
sustituimos w:
a1=2πn+asin⁡(w1)a_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}
a1=2πn+asin⁡(−1)a_{1} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}
a1=2πn−π2a_{1} = 2 \pi n - \frac{\pi}{2}
a2=2πn+asin⁡(w2)a_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}
a2=2πn+asin⁡(1)a_{2} = 2 \pi n + \operatorname{asin}{\left(1 \right)}
a2=2πn+π2a_{2} = 2 \pi n + \frac{\pi}{2}
a3=2πn−asin⁡(w1)+πa_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
a3=2πn−asin⁡(−1)+πa_{3} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi
a3=2πn+3π2a_{3} = 2 \pi n + \frac{3 \pi}{2}
a4=2πn−asin⁡(w2)+πa_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi
a4=2πn−asin⁡(1)+πa_{4} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi
a4=2πn+π2a_{4} = 2 \pi n + \frac{\pi}{2}
Gráfica
0-80-60-40-2020406080-10010004
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
  pi   pi
- -- + --
  2    2
−π2+π2- \frac{\pi}{2} + \frac{\pi}{2} 
=
0
00 
producto
-pi  pi
----*--
 2   2
−π2π2- \frac{\pi}{2} \frac{\pi}{2} 
=
   2 
-pi  
-----
  4
−π24- \frac{\pi^{2}}{4} 
-pi^2/4
Respuesta rápida [src] 
     -pi 
a1 = ----
      2
a1=−π2a_{1} = - \frac{\pi}{2} 
     pi
a2 = --
     2
a2=π2a_{2} = \frac{\pi}{2} 
a2 = pi/2
Respuesta numérica [src] 
a1 = 10.9955743696636
a2 = 39.2699084246933
a3 = -17.2787598091171
a4 = 54.9778711883962
a5 = -26.7035372990183
a6 = 48.6946859238715
a7 = 92.6769830795146
a8 = 1.5707965454425
a9 = 32.986722928111
a10 = 17.2787598502655
a11 = 4.71238876848081
a12 = 76.9690197631883
a13 = 39.2699081179815
a14 = -54.9778713137198
a15 = -86.393797765473
a16 = -14.1371668392726
a17 = 98.9601685932308
a18 = 26.7035373461441
a19 = -76.9690202568697
a20 = 10.9955740392793
a21 = -32.9867227513827
a22 = 17.2787595624179
a23 = 83.2522055730903
a24 = 70.6858345016621
a25 = -4.7123889912442
a26 = -42.4115006098842
a27 = -98.9601684414698
a28 = 51.8362788999928
a29 = 61.2610566752601
a30 = -51.8362786897497
a31 = -48.6946860920117
a32 = -61.2610569641117
a33 = -83.2522055415057
a34 = 83.2522052340866
a35 = -98.96016883042
a36 = -92.6769830239371
a37 = -39.2699083866483
a38 = 32.9867226137576
a39 = 64.4026493086922
a40 = 42.4115007291722
a41 = -67.5442421675773
a42 = -20.4203520321877
a43 = 61.2610569989704
a44 = 7.85398174058521
a45 = 80.1106126771746
a46 = -58.1194639993376
a47 = -92.6769831823972
a48 = 36.1283156002139
a49 = 541.924732890135
a50 = 80.1106131434937
a51 = -61.2610562242523
a52 = 54.9778714849733
a53 = -89.5353907467661
a54 = -76.9690198771149
a55 = 14.1371671048484
a56 = 45.553093700501
a57 = -73.8274272800405
a58 = -48.6946858738636
a59 = -23.5619450090417
a60 = -10.9955741902138
a61 = 20.4203521497111
a62 = -29.8451300963672
a63 = 58.1194644379895
a64 = -32.9867231091652
a65 = -95.8185758681287
a66 = 86.393797888273
a67 = 23.5619451230057
a68 = -7.85398149857354
a69 = -64.4026491876462
a70 = -70.685834448838
a71 = -98.960168684456
a72 = -10.9955745350309
a73 = 23.5619449395428
a74 = -26.7035375427973
a75 = 98.9601683381274
a76 = -39.2699081528781
a77 = -45.5530935883361
a78 = -36.1283154192437
a79 = -80.1106125795659
a80 = -17.2787590276524
a81 = 73.8274274795554
a82 = 67.5442422779275
a83 = -4.71238872430683
a84 = 76.9690200400775
a85 = -70.6858346386357
a86 = -54.9778716831146
a87 = 29.845130320338
a88 = 89.5353908552844
a89 = -1.57079642969308
a90 = 76.9690207492347
a91 = 95.8185760590309
a91 = 95.8185760590309
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