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Expresiones idénticas
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Expresiones semejantes
- 6sin^2x+1=0
6sin^2x-1=0 la ecuación
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Solución
Solución detallada
Tenemos la ecuación
$$6 \sin^{2}{\left(x \right)} - 1 = 0$$
cambiamos
$$6 \sin^{2}{\left(x \right)} - 1 = 0$$
$$6 \sin^{2}{\left(x \right)} - 1 = 0$$
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
$$a = 6$$
$$b = 0$$
$$c = -1$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = \frac{\sqrt{6}}{6}$$
$$w_{2} = - \frac{\sqrt{6}}{6}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{6}}{6} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{6}}{6} \right)} + \pi$$
$$x_{4} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)} + \pi$$
$$6 \sin^{2}{\left(x \right)} - 1 = 0$$
cambiamos
$$6 \sin^{2}{\left(x \right)} - 1 = 0$$
$$6 \sin^{2}{\left(x \right)} - 1 = 0$$
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
$$a = 6$$
$$b = 0$$
$$c = -1$$
, entonces
D = b^2 - 4 * a * c =
(0)^2 - 4 * (6) * (-1) = 24
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = \frac{\sqrt{6}}{6}$$
$$w_{2} = - \frac{\sqrt{6}}{6}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{6}}{6} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{6}}{6} \right)} + \pi$$
$$x_{4} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)} + \pi$$
Respuesta rápida
[src]
/ ___\
|\/ 6 |
x1 = pi - asin|-----|
\ 6 /
$$x_{1} = \pi - \operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)}$$
/ ___\
|\/ 6 |
x2 = pi + asin|-----|
\ 6 /
$$x_{2} = \operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)} + \pi$$
/ ___\
|\/ 6 |
x3 = -asin|-----|
\ 6 /
$$x_{3} = - \operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)}$$
/ ___\
|\/ 6 |
x4 = asin|-----|
\ 6 /
$$x_{4} = \operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)}$$
x4 = asin(sqrt(6)/6)
Suma y producto de raíces
[src]
suma
/ ___\ / ___\ / ___\ / ___\
|\/ 6 | |\/ 6 | |\/ 6 | |\/ 6 |
pi - asin|-----| + pi + asin|-----| - asin|-----| + asin|-----|
\ 6 / \ 6 / \ 6 / \ 6 /
$$\operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)} + \left(- \operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)} + \left(\left(\pi - \operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)}\right) + \left(\operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)} + \pi\right)\right)\right)$$
=
2*pi
$$2 \pi$$
producto
/ / ___\\ / / ___\\ / / ___\\ / ___\ | |\/ 6 || | |\/ 6 || | |\/ 6 || |\/ 6 | |pi - asin|-----||*|pi + asin|-----||*|-asin|-----||*asin|-----| \ \ 6 // \ \ 6 // \ \ 6 // \ 6 /
$$\left(\pi - \operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)}\right) \left(\operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)} + \pi\right) \left(- \operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)}\right) \operatorname{asin}{\left(\frac{\sqrt{6}}{6} \right)}$$
=
/ ___\ / / ___\ \
2|\/ 6 | | 2|\/ 6 | 2|
asin |-----|*|asin |-----| - pi |
\ 6 / \ \ 6 / /
$$\left(- \pi^{2} + \operatorname{asin}^{2}{\left(\frac{\sqrt{6}}{6} \right)}\right) \operatorname{asin}^{2}{\left(\frac{\sqrt{6}}{6} \right)}$$
asin(sqrt(6)/6)^2*(asin(sqrt(6)/6)^2 - pi^2)
Respuesta numérica
[src]
x1 = 3.56212698887376
x2 = -28.6948682175921
x3 = 56.1281334293323
x4 = 71.8360966972813
x5 = -93.8272452724098
x6 = 47.5444241391309
x7 = -43.5617628149731
x8 = 34.1369848542038
x9 = 5.86265097189562
x10 = 75.818758021439
x11 = -82.1019433286186
x12 = 9.84531229605335
x13 = 93.8272452724098
x14 = -21.5706142398446
x15 = -19.2700902568227
x16 = 40.4201701613833
x17 = 100.110430579589
x18 = 91.526721289388
x19 = -27.8537995470242
x20 = 2.72105831830583
x21 = -41.2612388319513
x22 = 24.7122068934344
x23 = 88.3851286357982
x24 = -87.5440599652302
x25 = -24.7122068934344
x26 = -78.1192820044609
x27 = -56.1281334293323
x28 = -40.4201701613833
x29 = -71.8360966972813
x30 = -46.7033554685629
x31 = -34.1369848542038
x32 = 78.1192820044609
x33 = -90.68565261882
x34 = -84.4024673116405
x35 = 82.1019433286186
x36 = 18.4290215862548
x37 = 62.4113187365119
x38 = -53.8276094463104
x39 = 90.68565261882
x40 = 69.5355727142594
x41 = -68.6945040436915
x42 = 74.9776893508711
x43 = -9.84531229605335
x44 = 97.8099065965676
x45 = -63.2523874070798
x46 = -38.1196461783615
x47 = -62.4113187365119
x48 = -97.8099065965676
x49 = 0.420534335283965
x50 = 46.7033554685629
x51 = -69.5355727142594
x52 = -2.72105831830583
x53 = 19.2700902568227
x54 = 66.3939800606696
x55 = -100.110430579589
x56 = 68.6945040436915
x57 = 192.057686204261
x58 = 53.8276094463104
x59 = -103.252023233179
x60 = 27.8537995470242
x61 = -3.56212698887376
x62 = 84.4024673116405
x63 = -5.86265097189562
x64 = -31.8364608711819
x65 = 31.8364608711819
x66 = -49.8449481221527
x67 = -18.4290215862548
x68 = -85.2435359822084
x69 = -60.11079475349
x70 = 44.4028314855411
x71 = 60.11079475349
x72 = 25.5532755640023
x73 = -47.5444241391309
x74 = -75.818758021439
x75 = 49.8449481221527
x76 = -16.1284976032329
x77 = -91.526721289388
x78 = -25.5532755640023
x79 = -65.5529113901017
x80 = 12.1458362790752
x81 = 16.1284976032329
x82 = -12.1458362790752
x83 = 38.1196461783615
x84 = 96.9688379259996
x85 = 22.4116829104125
x85 = 22.4116829104125