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Expresiones idénticas
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Expresiones semejantes
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Expresiones con funciones
- Seno sin
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- sin(x)+sin(2*x)+sin(3*x)=0
- Seno sin
- sin(x)=pi
- sin(4*x)=0
- sin(x)^2=1
- sin(x)=(1/2)
- sin(x)+sin(2*x)+sin(3*x)=0
sin(x)^(2)+(5/2)*sin(x)+1=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 5*sin(x)
sin (x) + -------- + 1 = 0
2
$$\left(\sin^{2}{\left(x \right)} + \frac{5 \sin{\left(x \right)}}{2}\right) + 1 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(\sin^{2}{\left(x \right)} + \frac{5 \sin{\left(x \right)}}{2}\right) + 1 = 0$$
cambiamos
$$\sin^{2}{\left(x \right)} + \frac{5 \sin{\left(x \right)}}{2} + 1 = 0$$
$$\left(\sin^{2}{\left(x \right)} + \frac{5 \sin{\left(x \right)}}{2}\right) + 1 = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
Es la ecuación de la forma
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
$$a = 1$$
$$b = \frac{5}{2}$$
$$c = 1$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = - \frac{1}{2}$$
$$w_{2} = -2$$
hacemos cambio inverso
$$\sin{\left(x \right)} = w$$
Tenemos la ecuación
$$\sin{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(-2 \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(2 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{7 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(-2 \right)}$$
$$x_{4} = 2 \pi n + \pi + \operatorname{asin}{\left(2 \right)}$$
$$\left(\sin^{2}{\left(x \right)} + \frac{5 \sin{\left(x \right)}}{2}\right) + 1 = 0$$
cambiamos
$$\sin^{2}{\left(x \right)} + \frac{5 \sin{\left(x \right)}}{2} + 1 = 0$$
$$\left(\sin^{2}{\left(x \right)} + \frac{5 \sin{\left(x \right)}}{2}\right) + 1 = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
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:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = \frac{5}{2}$$
$$c = 1$$
, entonces
D = b^2 - 4 * a * c =
(5/2)^2 - 4 * (1) * (1) = 9/4
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = - \frac{1}{2}$$
$$w_{2} = -2$$
hacemos cambio inverso
$$\sin{\left(x \right)} = w$$
Tenemos la ecuación
$$\sin{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(-2 \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(2 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{7 \pi}{6}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(-2 \right)}$$
$$x_{4} = 2 \pi n + \pi + \operatorname{asin}{\left(2 \right)}$$
Respuesta rápida
[src]
-pi
x1 = ----
6
$$x_{1} = - \frac{\pi}{6}$$
7*pi
x2 = ----
6
$$x_{2} = \frac{7 \pi}{6}$$
x3 = pi + I*im(asin(2)) + re(asin(2))
$$x_{3} = \operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}$$
x4 = -re(asin(2)) - I*im(asin(2))
$$x_{4} = - \operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}$$
x4 = -re(asin(2)) - i*im(asin(2))
Suma y producto de raíces
[src]
suma
pi 7*pi - -- + ---- + pi + I*im(asin(2)) + re(asin(2)) + -re(asin(2)) - I*im(asin(2)) 6 6
$$\left(\left(- \frac{\pi}{6} + \frac{7 \pi}{6}\right) + \left(\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right)\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right)$$
=
2*pi
$$2 \pi$$
producto
-pi 7*pi ----*----*(pi + I*im(asin(2)) + re(asin(2)))*(-re(asin(2)) - I*im(asin(2))) 6 6
$$- \frac{\pi}{6} \frac{7 \pi}{6} \left(\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right)$$
=
2
7*pi *(I*im(asin(2)) + re(asin(2)))*(pi + I*im(asin(2)) + re(asin(2)))
----------------------------------------------------------------------
36
$$\frac{7 \pi^{2} \left(\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(2 \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \right)}\right)}\right)}{36}$$
7*pi^2*(i*im(asin(2)) + re(asin(2)))*(pi + i*im(asin(2)) + re(asin(2)))/36
Respuesta numérica
[src]
x1 = 9.94837673636768
x2 = -75.9218224617533
x3 = 37.1755130674792
x4 = 74.8746249105567
x5 = 24.60914245312
x6 = 60.2138591938044
x7 = -15.1843644923507
x8 = -101.054563690472
x9 = -31.9395253114962
x10 = -25.6563400043166
x11 = 93.7241808320955
x12 = -52.8834763354282
x13 = 85.3466004225227
x14 = -0.523598775598299
x15 = -38.2227106186758
x16 = -57.0722665402146
x17 = -84.2994028713261
x18 = -90.5825881785057
x19 = 62.3082542961976
x20 = 66.497044500984
x21 = 16.2315620435473
x22 = 22.5147473507269
x23 = 28.7979326579064
x24 = 12.0427718387609
x25 = 238.237442897226
x26 = -46.6002910282486
x27 = -78.0162175641465
x28 = -82.2050077689329
x29 = 62766.4032248461
x30 = 100.007366139275
x31 = 5.75958653158129
x32 = -8.90117918517108
x33 = -13.0899693899575
x34 = -88.4881930761125
x35 = -94.7713783832921
x36 = -65.4498469497874
x37 = 79.0634151153431
x38 = -19.3731546971371
x39 = 56.025068989018
x40 = -34.0339204138894
x41 = -63.3554518473942
x42 = 91.6297857297023
x43 = -44.5058959258554
x44 = -6.80678408277789
x45 = -2.61799387799149
x46 = 72.7802298081635
x47 = 35.081117965086
x48 = -40.317105721069
x49 = 97.9129710368819
x50 = -96.8657734856853
x51 = 198.443935951755
x52 = 87.4409955249159
x53 = 81.1578102177363
x54 = -773.355391558687
x55 = 53.9306738866248
x56 = 68.5914396033772
x57 = 3.66519142918809
x58 = -71.733032256967
x59 = -50.789081233035
x60 = -59.1666616426078
x61 = 30.8923277602996
x62 = 47.6474885794452
x63 = -27.7507351067098
x64 = 18.3259571459405
x65 = 43.4586983746588
x66 = 41.3643032722656
x67 = -21.4675497995303
x68 = 49.7418836818384
x69 = -69.6386371545737
x69 = -69.6386371545737