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sin(a)^(2)=1/2 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
   2         
sin (a) = 1/2
$$\sin^{2}{\left(a \right)} = \frac{1}{2}$$
Solución detallada
Tenemos la ecuación
$$\sin^{2}{\left(a \right)} = \frac{1}{2}$$
cambiamos
$$- \frac{\cos{\left(2 a \right)}}{2} = 0$$
$$\sin^{2}{\left(a \right)} - \frac{1}{2} = 0$$
Sustituimos
$$w = \sin{\left(a \right)}$$
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:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = 0$$
$$c = - \frac{1}{2}$$
, entonces
D = b^2 - 4 * a * c = 

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

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

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

o
$$w_{1} = \frac{\sqrt{2}}{2}$$
$$w_{2} = - \frac{\sqrt{2}}{2}$$
hacemos cambio inverso
$$\sin{\left(a \right)} = w$$
Tenemos la ecuación
$$\sin{\left(a \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$a = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$a = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$a = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$a = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
sustituimos w:
$$a_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$a_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$a_{1} = 2 \pi n + \frac{\pi}{4}$$
$$a_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$a_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)}$$
$$a_{2} = 2 \pi n - \frac{\pi}{4}$$
$$a_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$a_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)} + \pi$$
$$a_{3} = 2 \pi n + \frac{3 \pi}{4}$$
$$a_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$a_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)} + \pi$$
$$a_{4} = 2 \pi n + \frac{5 \pi}{4}$$
Gráfica
Respuesta rápida [src]
     -pi 
a1 = ----
      4  
$$a_{1} = - \frac{\pi}{4}$$
     pi
a2 = --
     4 
$$a_{2} = \frac{\pi}{4}$$
     3*pi
a3 = ----
      4  
$$a_{3} = \frac{3 \pi}{4}$$
     5*pi
a4 = ----
      4  
$$a_{4} = \frac{5 \pi}{4}$$
a4 = 5*pi/4
Suma y producto de raíces [src]
suma
  pi   pi   3*pi   5*pi
- -- + -- + ---- + ----
  4    4     4      4  
$$\left(\left(- \frac{\pi}{4} + \frac{\pi}{4}\right) + \frac{3 \pi}{4}\right) + \frac{5 \pi}{4}$$
=
2*pi
$$2 \pi$$
producto
-pi  pi 3*pi 5*pi
----*--*----*----
 4   4   4    4  
$$\frac{5 \pi}{4} \frac{3 \pi}{4} \cdot - \frac{\pi}{4} \frac{\pi}{4}$$
=
      4
-15*pi 
-------
  256  
$$- \frac{15 \pi^{4}}{256}$$
-15*pi^4/256
Respuesta numérica [src]
a1 = -5.49778714378214
a2 = -38.484510006475
a3 = -40.0553063332699
a4 = 30.6305283725005
a5 = 84.037603483527
a6 = 85.6083998103219
a7 = 54.1924732744239
a8 = -68.329640215578
a9 = 19.6349540849362
a10 = 11.7809724509617
a11 = 68.329640215578
a12 = 44.7676953136546
a13 = -54.1924732744239
a14 = -47.9092879672443
a15 = -18.0641577581413
a16 = -16.4933614313464
a17 = 90.3207887907066
a18 = 8.63937979737193
a19 = -84.037603483527
a20 = -69.9004365423729
a21 = -19.6349540849362
a22 = 49.4800842940392
a23 = -27.4889357189107
a24 = -3.92699081698724
a25 = 41.6261026600648
a26 = -55.7632696012188
a27 = -76.1836218495525
a28 = 62.0464549083984
a29 = -32.2013246992954
a30 = -46.3384916404494
a31 = 25.9181393921158
a32 = -77.7544181763474
a33 = 47.9092879672443
a34 = 91.8915851175014
a35 = 24.3473430653209
a36 = 38.484510006475
a37 = 99.7455667514759
a38 = 40.0553063332699
a39 = 66.7588438887831
a40 = 384.059701901352
a41 = -13.3517687777566
a42 = 88.7499924639117
a43 = 98.174770424681
a44 = 10.2101761241668
a45 = -90.3207887907066
a46 = 55.7632696012188
a47 = -49.4800842940392
a48 = 22.776546738526
a49 = -79.3252145031423
a50 = 60.4756585816035
a51 = 74.6128255227576
a52 = -99.7455667514759
a53 = -24.3473430653209
a54 = -71.4712328691678
a55 = 76.1836218495525
a56 = 3.92699081698724
a57 = -62.0464549083984
a58 = -33.7721210260903
a59 = 18.0641577581413
a60 = -41.6261026600648
a61 = 162.577419823272
a62 = -85.6083998103219
a63 = -35.3429173528852
a64 = 87.1791961371168
a65 = 52.621676947629
a66 = 69.9004365423729
a67 = 96.6039740978861
a68 = -82.4668071567321
a69 = -63.6172512351933
a70 = -11.7809724509617
a71 = 27.4889357189107
a72 = -10.2101761241668
a73 = 82.4668071567321
a74 = 46.3384916404494
a75 = -1144.32512407008
a76 = -93.4623814442964
a77 = -60.4756585816035
a78 = -91.8915851175014
a79 = 32.2013246992954
a80 = -98.174770424681
a81 = -2.35619449019234
a82 = 63.6172512351933
a83 = 5.49778714378214
a84 = -57.3340659280137
a85 = 77.7544181763474
a86 = -25.9181393921158
a87 = 16.4933614313464
a88 = 2.35619449019234
a89 = 33.7721210260903
a89 = 33.7721210260903