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Expresiones semejantes
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Expresiones con funciones
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-2*sin(x)+cos(x)^2=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 -2*sin(x) + cos (x) = 0
$$- 2 \sin{\left(x \right)} + \cos^{2}{\left(x \right)} = 0$$
Solución detallada
Tenemos la ecuación
$$- 2 \sin{\left(x \right)} + \cos^{2}{\left(x \right)} = 0$$
cambiamos
$$- 2 \sin{\left(x \right)} + \cos^{2}{\left(x \right)} = 0$$
$$- \sin^{2}{\left(x \right)} - 2 \sin{\left(x \right)} + 1 = 0$$
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
$$a = -1$$
$$b = -2$$
$$c = 1$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = - \sqrt{2} - 1$$
$$w_{2} = -1 + \sqrt{2}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \sqrt{2} - 1 \right)}$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(1 + \sqrt{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(-1 + \sqrt{2} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(1 - \sqrt{2} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(- \sqrt{2} - 1 \right)}$$
$$x_{3} = 2 \pi n + \pi + \operatorname{asin}{\left(1 + \sqrt{2} \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(-1 + \sqrt{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \operatorname{asin}{\left(1 - \sqrt{2} \right)} + \pi$$
$$- 2 \sin{\left(x \right)} + \cos^{2}{\left(x \right)} = 0$$
cambiamos
$$- 2 \sin{\left(x \right)} + \cos^{2}{\left(x \right)} = 0$$
$$- \sin^{2}{\left(x \right)} - 2 \sin{\left(x \right)} + 1 = 0$$
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
$$a = -1$$
$$b = -2$$
$$c = 1$$
, entonces
D = b^2 - 4 * a * c =
(-2)^2 - 4 * (-1) * (1) = 8
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = - \sqrt{2} - 1$$
$$w_{2} = -1 + \sqrt{2}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \sqrt{2} - 1 \right)}$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(1 + \sqrt{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(-1 + \sqrt{2} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(1 - \sqrt{2} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(- \sqrt{2} - 1 \right)}$$
$$x_{3} = 2 \pi n + \pi + \operatorname{asin}{\left(1 + \sqrt{2} \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(-1 + \sqrt{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \operatorname{asin}{\left(1 - \sqrt{2} \right)} + \pi$$
Suma y producto de raíces
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suma
/ / ___________\\ / / ___________\\ / ___________\ / / ___________\\ / / ___________\\ / ___________\
| | ___ ___ / ___ || | | ___ ___ / ___ || | ___ ___ / ___ | | | ___ ___ / ___ || | | ___ ___ / ___ || | ___ ___ / ___ |
- 2*re\atan\-1 + \/ 2 + \/ 2 *\/ 1 - \/ 2 // - 2*I*im\atan\-1 + \/ 2 + \/ 2 *\/ 1 - \/ 2 // + 2*atan\1 + \/ 2 + \/ 2 *\/ 1 + \/ 2 / + 2*re\atan\1 - \/ 2 + \/ 2 *\/ 1 - \/ 2 // + 2*I*im\atan\1 - \/ 2 + \/ 2 *\/ 1 - \/ 2 // + 2*atan\1 + \/ 2 - \/ 2 *\/ 1 + \/ 2 /
$$\left(\left(2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{1 + \sqrt{2}} \right)} + \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(-1 + \sqrt{2} + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(-1 + \sqrt{2} + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)}\right)\right) + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)}\right)\right) + 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{1 + \sqrt{2}} + 1 + \sqrt{2} \right)}$$
=
/ / ___________\\ / ___________\ / ___________\ / / ___________\\ / / ___________\\ / / ___________\\
| | ___ ___ / ___ || | ___ ___ / ___ | | ___ ___ / ___ | | | ___ ___ / ___ || | | ___ ___ / ___ || | | ___ ___ / ___ ||
- 2*re\atan\-1 + \/ 2 + \/ 2 *\/ 1 - \/ 2 // + 2*atan\1 + \/ 2 + \/ 2 *\/ 1 + \/ 2 / + 2*atan\1 + \/ 2 - \/ 2 *\/ 1 + \/ 2 / + 2*re\atan\1 - \/ 2 + \/ 2 *\/ 1 - \/ 2 // - 2*I*im\atan\-1 + \/ 2 + \/ 2 *\/ 1 - \/ 2 // + 2*I*im\atan\1 - \/ 2 + \/ 2 *\/ 1 - \/ 2 //
$$- 2 \operatorname{re}{\left(\operatorname{atan}{\left(-1 + \sqrt{2} + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} + 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{1 + \sqrt{2}} + 1 + \sqrt{2} \right)} + 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{1 + \sqrt{2}} \right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(-1 + \sqrt{2} + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)}$$
producto
/ / / ___________\\ / / ___________\\\ / ___________\ / / / ___________\\ / / ___________\\\ / ___________\ | | | ___ ___ / ___ || | | ___ ___ / ___ ||| | ___ ___ / ___ | | | | ___ ___ / ___ || | | ___ ___ / ___ ||| | ___ ___ / ___ | \- 2*re\atan\-1 + \/ 2 + \/ 2 *\/ 1 - \/ 2 // - 2*I*im\atan\-1 + \/ 2 + \/ 2 *\/ 1 - \/ 2 ///*2*atan\1 + \/ 2 + \/ 2 *\/ 1 + \/ 2 /*\2*re\atan\1 - \/ 2 + \/ 2 *\/ 1 - \/ 2 // + 2*I*im\atan\1 - \/ 2 + \/ 2 *\/ 1 - \/ 2 ///*2*atan\1 + \/ 2 - \/ 2 *\/ 1 + \/ 2 /
$$\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(-1 + \sqrt{2} + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(-1 + \sqrt{2} + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)}\right) 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{1 + \sqrt{2}} \right)} \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)}\right) 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{1 + \sqrt{2}} + 1 + \sqrt{2} \right)}$$
=
/ / / ___________\\ / / ___________\\\ / / / ___________\\ / / ___________\\\ / ___________\ / ___________\
| | | ___ ___ / ___ || | | ___ ___ / ___ ||| | | | ___ ___ / ___ || | | ___ ___ / ___ ||| | ___ ___ / ___ | | ___ ___ / ___ |
-16*\I*im\atan\1 - \/ 2 + \/ 2 *\/ 1 - \/ 2 // + re\atan\1 - \/ 2 + \/ 2 *\/ 1 - \/ 2 ///*\I*im\atan\-1 + \/ 2 + \/ 2 *\/ 1 - \/ 2 // + re\atan\-1 + \/ 2 + \/ 2 *\/ 1 - \/ 2 ///*atan\1 + \/ 2 + \/ 2 *\/ 1 + \/ 2 /*atan\1 + \/ 2 - \/ 2 *\/ 1 + \/ 2 /
$$- 16 \left(\operatorname{re}{\left(\operatorname{atan}{\left(-1 + \sqrt{2} + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(-1 + \sqrt{2} + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)}\right) \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{1 + \sqrt{2}} \right)} \operatorname{atan}{\left(- \sqrt{2} \sqrt{1 + \sqrt{2}} + 1 + \sqrt{2} \right)}$$
-16*(i*im(atan(1 - sqrt(2) + sqrt(2)*sqrt(1 - sqrt(2)))) + re(atan(1 - sqrt(2) + sqrt(2)*sqrt(1 - sqrt(2)))))*(i*im(atan(-1 + sqrt(2) + sqrt(2)*sqrt(1 - sqrt(2)))) + re(atan(-1 + sqrt(2) + sqrt(2)*sqrt(1 - sqrt(2)))))*atan(1 + sqrt(2) + sqrt(2)*sqrt(1 + sqrt(2)))*atan(1 + sqrt(2) - sqrt(2)*sqrt(1 + sqrt(2)))
Respuesta rápida
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/ / ___________\\ / / ___________\\
| | ___ ___ / ___ || | | ___ ___ / ___ ||
x1 = - 2*re\atan\-1 + \/ 2 + \/ 2 *\/ 1 - \/ 2 // - 2*I*im\atan\-1 + \/ 2 + \/ 2 *\/ 1 - \/ 2 //
$$x_{1} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(-1 + \sqrt{2} + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(-1 + \sqrt{2} + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)}$$
/ ___________\
| ___ ___ / ___ |
x2 = 2*atan\1 + \/ 2 + \/ 2 *\/ 1 + \/ 2 /
$$x_{2} = 2 \operatorname{atan}{\left(1 + \sqrt{2} + \sqrt{2} \sqrt{1 + \sqrt{2}} \right)}$$
/ / ___________\\ / / ___________\\
| | ___ ___ / ___ || | | ___ ___ / ___ ||
x3 = 2*re\atan\1 - \/ 2 + \/ 2 *\/ 1 - \/ 2 // + 2*I*im\atan\1 - \/ 2 + \/ 2 *\/ 1 - \/ 2 //
$$x_{3} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \sqrt{2} + 1 + \sqrt{2} \sqrt{1 - \sqrt{2}} \right)}\right)}$$
/ ___________\
| ___ ___ / ___ |
x4 = 2*atan\1 + \/ 2 - \/ 2 *\/ 1 + \/ 2 /
$$x_{4} = 2 \operatorname{atan}{\left(- \sqrt{2} \sqrt{1 + \sqrt{2}} + 1 + \sqrt{2} \right)}$$
x4 = 2*atan(-sqrt(2)*sqrt(1 + sqrt(2)) + 1 + sqrt(2))
Respuesta numérica
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x1 = -16.1350418543414
x2 = 19.2766345079312
x3 = 6.71026389357206
x4 = 94.6748581940863
x5 = -22.418227161521
x6 = 21.5640699887361
x7 = 100.958043501266
x8 = 63.2589316581883
x9 = 90.6791083677115
x10 = 84.3959230605319
x11 = -91.5332655404965
x12 = 12.9934492007516
x13 = 44.4093757366496
x14 = 34.1304406030953
x15 = -53.834153697419
x16 = 71.8295524461728
x17 = -30.9888479495055
x18 = -28.7014124687006
x19 = 59.2631818318136
x20 = 0.427078586392476
x21 = -34.9845977758802
x22 = 78.1127377533523
x23 = 46.6968112174544
x24 = -122.949192076394
x25 = -72.6837096189577
x26 = 96.9622936748911
x27 = -56.1215891782238
x28 = -47.5509683902394
x29 = -85.2500802333169
x30 = 38.12619042947
x31 = 27.8472552959157
x32 = -631.887201957941
x33 = 15.2808846815565
x34 = -68.687959792583
x35 = -49.8384038710442
x36 = -37.272033256685
x37 = -12.1392920279667
x38 = 75.8253022725475
x39 = -74.9711450997626
x40 = 8.9976993743769
x41 = -87.5375157141217
x42 = -100.103886328481
x43 = 2.71451406719732
x44 = -110.382821462035
x45 = 65.5463671389932
x46 = -18.4224773351463
x47 = 82.1084875797271
x48 = -43.5552185638646
x49 = -9.85185654716186
x50 = 88.3916728869067
x51 = 56.9757463510088
x52 = 40.4136259102748
x53 = -66.4005243117781
x54 = -78.9668949261373
x55 = -81.2543304069421
x56 = -97.8164508476761
x57 = -93.8207010213013
x58 = -41.2677830830598
x59 = -3.56867123998227
x60 = 69.5421169653679
x61 = 31.8430051222904
x62 = 50.6925610438292
x63 = -60.1173390045985
x64 = -5.85610672078711
x65 = -62.4047744854034
x66 = 25.5598198151108
x67 = 52.979996524634
x68 = -24.7056626423259
x68 = -24.7056626423259