Sr Examen

Otras calculadoras

tg^2x=1/2 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

v

Solución numérica:

Buscar la solución numérica en el intervalo [, ]

Solución

Ha introducido [src]
   2         
tan (x) = 1/2
$$\tan^{2}{\left(x \right)} = \frac{1}{2}$$
Solución detallada
Tenemos la ecuación
$$\tan^{2}{\left(x \right)} = \frac{1}{2}$$
cambiamos
$$\tan^{2}{\left(x \right)} - \frac{1}{2} = 0$$
$$\tan^{2}{\left(x \right)} - \frac{1}{2} = 0$$
Sustituimos
$$w = \tan{\left(x \right)}$$
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:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
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
$$\tan{\left(x \right)} = w$$
Tenemos la ecuación
$$\tan{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
O
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = \pi n + \operatorname{atan}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{atan}{\left(- \frac{\sqrt{2}}{2} \right)}$$
$$x_{2} = \pi n - \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}$$
Gráfica
Suma y producto de raíces [src]
suma
      /  ___\       /  ___\
      |\/ 2 |       |\/ 2 |
- atan|-----| + atan|-----|
      \  2  /       \  2  /
$$- \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)} + \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}$$
=
0
$$0$$
producto
     /  ___\     /  ___\
     |\/ 2 |     |\/ 2 |
-atan|-----|*atan|-----|
     \  2  /     \  2  /
$$- \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)} \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}$$
=
      /  ___\
     2|\/ 2 |
-atan |-----|
      \  2  /
$$- \operatorname{atan}^{2}{\left(\frac{\sqrt{2}}{2} \right)}$$
-atan(sqrt(2)/2)^2
Respuesta rápida [src]
          /  ___\
          |\/ 2 |
x1 = -atan|-----|
          \  2  /
$$x_{1} = - \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}$$
         /  ___\
         |\/ 2 |
x2 = atan|-----|
         \  2  /
$$x_{2} = \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}$$
x2 = atan(sqrt(2)/2)
Respuesta numérica [src]
x1 = -85.4384813555948
x2 = -93.6322998990234
x3 = 82.296888702005
x4 = -60.3057401268765
x5 = 69.7305180876458
x6 = 44.5977768589275
x7 = 8.80929825209899
x8 = 52.7915954023561
x9 = 40.2252247879969
x10 = -46.5084100951765
x11 = -79.1552960484152
x12 = 84.207521938254
x13 = -91.7216666627744
x14 = 2.52611294491941
x15 = -8.80929825209899
x16 = 11.9508909056888
x17 = 74.7827439774847
x18 = 96.7738925526132
x19 = -98.004851969954
x20 = 16.3234429766194
x21 = -49.6500027487663
x22 = -90.4907072454336
x23 = -55.9331880559459
x24 = 68.4995586703051
x25 = -4791.5442764331
x26 = -47.7393695125173
x27 = -21.3756688664582
x28 = 38.3145915517479
x29 = 32.0314062445683
x30 = 55.9331880559459
x31 = -82.296888702005
x32 = -27.6588541736378
x33 = 33.9420394808173
x34 = -132.562371159442
x35 = -71.6411513238949
x36 = 81.0659292846642
x37 = 63.4473327804663
x38 = -74.7827439774847
x39 = -10.0402576694398
x40 = 85.4384813555948
x41 = 62.2163733631255
x42 = 19.4650356302091
x43 = -30.8004468272275
x44 = 71.6411513238949
x45 = 46.5084100951765
x46 = -13.1818503230296
x47 = -24.517261520048
x48 = 76.0137033948254
x49 = 0.615479708670387
x50 = 99.915485206203
x51 = 18.2340762128684
x52 = 3.75707236226018
x53 = 90.4907072454336
x54 = 25.7482209373887
x55 = -25.7482209373887
x56 = 91.7216666627744
x57 = 54.0225548196969
x58 = -41.4561842053377
x59 = -35.1729988981581
x60 = -19.4650356302091
x61 = -2.52611294491941
x62 = 98.004851969954
x63 = -57.1641474732867
x64 = 30.8004468272275
x65 = -76.0137033948254
x66 = -16.3234429766194
x67 = 59.0747807095357
x68 = -52.7915954023561
x69 = -40.2252247879969
x70 = -18.2340762128684
x71 = -11.9508909056888
x72 = 22.6066282837989
x73 = -77.9243366310744
x74 = -68.4995586703051
x75 = -32.0314062445683
x76 = -63.4473327804663
x77 = 88.5800740091846
x78 = 47.7393695125173
x79 = -99.915485206203
x80 = 66.5889254340561
x81 = 77.9243366310744
x82 = -84.207521938254
x83 = -96.7738925526132
x84 = -5.6677055985092
x85 = 41.4561842053377
x86 = -62.2163733631255
x87 = -69.7305180876458
x88 = 10.0402576694398
x89 = 13.1818503230296
x90 = 93.6322998990234
x91 = -54.0225548196969
x92 = 24.517261520048
x93 = -33.9420394808173
x94 = -3.75707236226018
x95 = 60.3057401268765
x95 = 60.3057401268765