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sin(2*x-pi*1/4)=1/2

sin(2*x-pi*1/4)=1/2 la ecuación

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Solución numérica:

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Solución

Ha introducido [src]
   /      pi\      
sin|2*x - --| = 1/2
   \      4 /      
$$\sin{\left(2 x - \frac{\pi}{4} \right)} = \frac{1}{2}$$
Solución detallada
Tenemos la ecuación
$$\sin{\left(2 x - \frac{\pi}{4} \right)} = \frac{1}{2}$$
es la ecuación trigonométrica más simple
Dividamos ambos miembros de la ecuación en -1

La ecuación se convierte en
$$\cos{\left(2 x + \frac{\pi}{4} \right)} = - \frac{1}{2}$$
Esta ecuación se reorganiza en
$$2 x + \frac{\pi}{4} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$2 x + \frac{\pi}{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
O
$$2 x + \frac{\pi}{4} = \pi n + \frac{2 \pi}{3}$$
$$2 x + \frac{\pi}{4} = \pi n - \frac{\pi}{3}$$
, donde n es cualquier número entero
Transportemos
$$\frac{\pi}{4}$$
al miembro derecho de la ecuación
con el signo opuesto, en total:
$$2 x = \pi n + \frac{5 \pi}{12}$$
$$2 x = \pi n - \frac{7 \pi}{12}$$
Dividamos ambos miembros de la ecuación obtenida en
$$2$$
obtenemos la respuesta:
$$x_{1} = \frac{\pi n}{2} + \frac{5 \pi}{24}$$
$$x_{2} = \frac{\pi n}{2} - \frac{7 \pi}{24}$$
Gráfica
Respuesta rápida [src]
     5*pi
x1 = ----
      24 
$$x_{1} = \frac{5 \pi}{24}$$
     13*pi
x2 = -----
       24 
$$x_{2} = \frac{13 \pi}{24}$$
x2 = 13*pi/24
Suma y producto de raíces [src]
suma
5*pi   13*pi
---- + -----
 24      24 
$$\frac{5 \pi}{24} + \frac{13 \pi}{24}$$
=
3*pi
----
 4  
$$\frac{3 \pi}{4}$$
producto
5*pi 13*pi
----*-----
 24    24 
$$\frac{5 \pi}{24} \frac{13 \pi}{24}$$
=
     2
65*pi 
------
 576  
$$\frac{65 \pi^{2}}{576}$$
65*pi^2/576
Respuesta numérica [src]
x1 = -70.5549350118708
x2 = -29.7142305152035
x3 = 58.2503637853107
x4 = -93.5932811381959
x5 = -35.997415822383
x6 = 7.98488132787406
x7 = -1.43989663289532
x8 = -14.0062672472545
x9 = -48.5637864367422
x10 = 94.9022780771917
x11 = 64.5335490924903
x12 = -49.6109839879388
x13 = -11.9118721448613
x14 = -87.3100958310163
x15 = -84.1685031774265
x16 = 20.5512519422332
x17 = 80.2415123604393
x18 = -57.9885643975116
x19 = 0.654498469497874
x20 = -67.413342358281
x21 = 45.6839931709516
x22 = 17.4096592886434
x23 = 60.3447588877039
x24 = -55.8941692951184
x25 = 23.692844595823
x26 = 95.9494756283883
x27 = 32.0704250053958
x28 = 4318.25000205307
x29 = -4.58148928648512
x30 = -51.705379090332
x31 = -33.9030207199899
x32 = -21.3366501056307
x33 = -40.1862060271694
x34 = 89.6662903212087
x35 = 98.0438707307815
x36 = -71.6021325630674
x37 = 3.79609112308767
x38 = 10.0792764302673
x39 = 44.636795619755
x40 = 47.7783882733448
x41 = 61.3919564389005
x42 = 86.5246976676189
x43 = -62.177354602298
x44 = 82.3359074628325
x45 = -64.2717497046912
x46 = 42.5424005173618
x47 = 39.400807863772
x48 = -42.2806011295626
x49 = -95.6876762405891
x50 = -7.72308194007491
x51 = 76.0527221556529
x52 = -77.885317870247
x53 = 66.6279441948835
x54 = 72.9111295020631
x55 = 11.1264739814639
x56 = 14.2680666350536
x57 = 36.2592152101822
x58 = -86.2628982798197
x59 = 25.7872396982162
x60 = 22.6456470446264
x61 = -45.4221937831524
x62 = -18.1950574520409
x63 = -43.3277986807592
x64 = 4.84328867428426
x65 = 54.0615735805244
x66 = -27.6198354128103
x67 = 88.6190927700121
x68 = 67.6751417460801
x69 = -20.2894525544341
x70 = -73.6965276654606
x71 = 91.7606854236019
x72 = 73.9583270532597
x73 = 38.3536103125754
x74 = -23.4310452080239
x75 = -99.8764664453755
x76 = -26.5726378616137
x77 = 16.3624617374468
x78 = 69.7695368484733
x79 = 50.9199809269346
x80 = -65.3189472558878
x81 = 26.8344372494128
x82 = -101.970861547769
x83 = -5.62868683768171
x84 = -92.5460835869993
x85 = 83.3831050140291
x86 = 29.9760299030026
x87 = -79.9797129726402
x88 = 1.70169602069447
x89 = 51.9671784781312
x89 = 51.9671784781312
Gráfico
sin(2*x-pi*1/4)=1/2 la ecuación