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Expresiones idénticas
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- sin(x)+2cos2(x)=0
- sinx+2cos2x=0
- sin(x)+2cos²(x)=0
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- sin(x)+2cos^2(x)=O
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Expresiones semejantes
- sin(x)-2cos^2(x)=0
- sinx+2cos^2(x)=0
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Expresiones con funciones
- Seno sin
- sin(sin(x))=1
- sin(-x)=1/2
- sin(z)=0
- sin(x-pi/3)+1=0
- sin(x+3*pi/4)=0
sin(x)+2cos^2(x)=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 sin(x) + 2*cos (x) = 0
$$\sin{\left(x \right)} + 2 \cos^{2}{\left(x \right)} = 0$$
Solución detallada
Tenemos la ecuación
$$\sin{\left(x \right)} + 2 \cos^{2}{\left(x \right)} = 0$$
cambiamos
$$\sin{\left(x \right)} + \cos{\left(2 x \right)} + 1 = 0$$
$$- 2 \sin^{2}{\left(x \right)} + \sin{\left(x \right)} + 2 = 0$$
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
$$a = -2$$
$$b = 1$$
$$c = 2$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = \frac{1}{4} - \frac{\sqrt{17}}{4}$$
$$w_{2} = \frac{1}{4} + \frac{\sqrt{17}}{4}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} - \frac{\sqrt{17}}{4} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} - \frac{\sqrt{17}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} + \frac{\sqrt{17}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} + \frac{\sqrt{17}}{4} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} - \frac{\sqrt{17}}{4} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} - \frac{\sqrt{17}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{1}{4} + \frac{\sqrt{17}}{4} \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{1}{4} + \frac{\sqrt{17}}{4} \right)}$$
$$\sin{\left(x \right)} + 2 \cos^{2}{\left(x \right)} = 0$$
cambiamos
$$\sin{\left(x \right)} + \cos{\left(2 x \right)} + 1 = 0$$
$$- 2 \sin^{2}{\left(x \right)} + \sin{\left(x \right)} + 2 = 0$$
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
$$a = -2$$
$$b = 1$$
$$c = 2$$
, entonces
D = b^2 - 4 * a * c =
(1)^2 - 4 * (-2) * (2) = 17
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = \frac{1}{4} - \frac{\sqrt{17}}{4}$$
$$w_{2} = \frac{1}{4} + \frac{\sqrt{17}}{4}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} - \frac{\sqrt{17}}{4} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} - \frac{\sqrt{17}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} + \frac{\sqrt{17}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} + \frac{\sqrt{17}}{4} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} - \frac{\sqrt{17}}{4} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} - \frac{\sqrt{17}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{1}{4} + \frac{\sqrt{17}}{4} \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{1}{4} + \frac{\sqrt{17}}{4} \right)}$$
Respuesta rápida
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/ / ____________\\ / / ____________\\
| | ____ ___ / ____ || | | ____ ___ / ____ ||
| | 1 \/ 17 \/ 2 *\/ 1 - \/ 17 || | | 1 \/ 17 \/ 2 *\/ 1 - \/ 17 ||
x1 = 2*re|atan|- - + ------ + ---------------------|| + 2*I*im|atan|- - + ------ + ---------------------||
\ \ 4 4 4 // \ \ 4 4 4 //
$$x_{1} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{1 - \sqrt{17}}}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{1 - \sqrt{17}}}{4} \right)}\right)}$$
/ ____________\
| ____ ___ / ____ |
|1 \/ 17 \/ 2 *\/ 1 + \/ 17 |
x2 = -2*atan|- + ------ + ---------------------|
\4 4 4 /
$$x_{2} = - 2 \operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{2} \sqrt{1 + \sqrt{17}}}{4} + \frac{\sqrt{17}}{4} \right)}$$
/ / ____________\\ / / ____________\\
| | ____ ___ / ____ || | | ____ ___ / ____ ||
| |1 \/ 17 \/ 2 *\/ 1 - \/ 17 || | |1 \/ 17 \/ 2 *\/ 1 - \/ 17 ||
x3 = - 2*re|atan|- - ------ + ---------------------|| - 2*I*im|atan|- - ------ + ---------------------||
\ \4 4 4 // \ \4 4 4 //
$$x_{3} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{17}}{4} + \frac{1}{4} + \frac{\sqrt{2} \sqrt{1 - \sqrt{17}}}{4} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{17}}{4} + \frac{1}{4} + \frac{\sqrt{2} \sqrt{1 - \sqrt{17}}}{4} \right)}\right)}$$
/ ____________\
| ____ ___ / ____ |
|1 \/ 17 \/ 2 *\/ 1 + \/ 17 |
x4 = -2*atan|- + ------ - ---------------------|
\4 4 4 /
$$x_{4} = - 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{1 + \sqrt{17}}}{4} + \frac{1}{4} + \frac{\sqrt{17}}{4} \right)}$$
x4 = -2*atan(-sqrt(2)*sqrt(1 + sqrt(17))/4 + 1/4 + sqrt(17)/4)
Suma y producto de raíces
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suma
/ / ____________\\ / / ____________\\ / ____________\ / / ____________\\ / / ____________\\ / ____________\
| | ____ ___ / ____ || | | ____ ___ / ____ || | ____ ___ / ____ | | | ____ ___ / ____ || | | ____ ___ / ____ || | ____ ___ / ____ |
| | 1 \/ 17 \/ 2 *\/ 1 - \/ 17 || | | 1 \/ 17 \/ 2 *\/ 1 - \/ 17 || |1 \/ 17 \/ 2 *\/ 1 + \/ 17 | | |1 \/ 17 \/ 2 *\/ 1 - \/ 17 || | |1 \/ 17 \/ 2 *\/ 1 - \/ 17 || |1 \/ 17 \/ 2 *\/ 1 + \/ 17 |
2*re|atan|- - + ------ + ---------------------|| + 2*I*im|atan|- - + ------ + ---------------------|| - 2*atan|- + ------ + ---------------------| + - 2*re|atan|- - ------ + ---------------------|| - 2*I*im|atan|- - ------ + ---------------------|| - 2*atan|- + ------ - ---------------------|
\ \ 4 4 4 // \ \ 4 4 4 // \4 4 4 / \ \4 4 4 // \ \4 4 4 // \4 4 4 /
$$- 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{1 + \sqrt{17}}}{4} + \frac{1}{4} + \frac{\sqrt{17}}{4} \right)} + \left(\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{17}}{4} + \frac{1}{4} + \frac{\sqrt{2} \sqrt{1 - \sqrt{17}}}{4} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{17}}{4} + \frac{1}{4} + \frac{\sqrt{2} \sqrt{1 - \sqrt{17}}}{4} \right)}\right)}\right) + \left(- 2 \operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{2} \sqrt{1 + \sqrt{17}}}{4} + \frac{\sqrt{17}}{4} \right)} + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{1 - \sqrt{17}}}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{1 - \sqrt{17}}}{4} \right)}\right)}\right)\right)\right)$$
=
/ ____________\ / ____________\ / / ____________\\ / / ____________\\ / / ____________\\ / / ____________\\
| ____ ___ / ____ | | ____ ___ / ____ | | | ____ ___ / ____ || | | ____ ___ / ____ || | | ____ ___ / ____ || | | ____ ___ / ____ ||
|1 \/ 17 \/ 2 *\/ 1 + \/ 17 | |1 \/ 17 \/ 2 *\/ 1 + \/ 17 | | |1 \/ 17 \/ 2 *\/ 1 - \/ 17 || | | 1 \/ 17 \/ 2 *\/ 1 - \/ 17 || | |1 \/ 17 \/ 2 *\/ 1 - \/ 17 || | | 1 \/ 17 \/ 2 *\/ 1 - \/ 17 ||
- 2*atan|- + ------ - ---------------------| - 2*atan|- + ------ + ---------------------| - 2*re|atan|- - ------ + ---------------------|| + 2*re|atan|- - + ------ + ---------------------|| - 2*I*im|atan|- - ------ + ---------------------|| + 2*I*im|atan|- - + ------ + ---------------------||
\4 4 4 / \4 4 4 / \ \4 4 4 // \ \ 4 4 4 // \ \4 4 4 // \ \ 4 4 4 //
$$- 2 \operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{2} \sqrt{1 + \sqrt{17}}}{4} + \frac{\sqrt{17}}{4} \right)} - 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{1 + \sqrt{17}}}{4} + \frac{1}{4} + \frac{\sqrt{17}}{4} \right)} - 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{17}}{4} + \frac{1}{4} + \frac{\sqrt{2} \sqrt{1 - \sqrt{17}}}{4} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{1 - \sqrt{17}}}{4} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{17}}{4} + \frac{1}{4} + \frac{\sqrt{2} \sqrt{1 - \sqrt{17}}}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{1 - \sqrt{17}}}{4} \right)}\right)}$$
producto
/ / / ____________\\ / / ____________\\\ / ____________\ / / / ____________\\ / / ____________\\\ / ____________\ | | | ____ ___ / ____ || | | ____ ___ / ____ ||| | ____ ___ / ____ | | | | ____ ___ / ____ || | | ____ ___ / ____ ||| | ____ ___ / ____ | | | | 1 \/ 17 \/ 2 *\/ 1 - \/ 17 || | | 1 \/ 17 \/ 2 *\/ 1 - \/ 17 ||| |1 \/ 17 \/ 2 *\/ 1 + \/ 17 | | | |1 \/ 17 \/ 2 *\/ 1 - \/ 17 || | |1 \/ 17 \/ 2 *\/ 1 - \/ 17 ||| |1 \/ 17 \/ 2 *\/ 1 + \/ 17 | |2*re|atan|- - + ------ + ---------------------|| + 2*I*im|atan|- - + ------ + ---------------------|||*-2*atan|- + ------ + ---------------------|*|- 2*re|atan|- - ------ + ---------------------|| - 2*I*im|atan|- - ------ + ---------------------|||*-2*atan|- + ------ - ---------------------| \ \ \ 4 4 4 // \ \ 4 4 4 /// \4 4 4 / \ \ \4 4 4 // \ \4 4 4 /// \4 4 4 /
$$\left(2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{1 - \sqrt{17}}}{4} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{1 - \sqrt{17}}}{4} \right)}\right)}\right) \left(- 2 \operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{2} \sqrt{1 + \sqrt{17}}}{4} + \frac{\sqrt{17}}{4} \right)}\right) \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{17}}{4} + \frac{1}{4} + \frac{\sqrt{2} \sqrt{1 - \sqrt{17}}}{4} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{17}}{4} + \frac{1}{4} + \frac{\sqrt{2} \sqrt{1 - \sqrt{17}}}{4} \right)}\right)}\right) \left(- 2 \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{1 + \sqrt{17}}}{4} + \frac{1}{4} + \frac{\sqrt{17}}{4} \right)}\right)$$
=
/ / / ____________\\ / / ____________\\\ / / / ____________\\ / / ____________\\\ / ____________\ / ____________\
| | | ____ ___ / ____ || | | ____ ___ / ____ ||| | | | ____ ___ / ____ || | | ____ ___ / ____ ||| | ____ ___ / ____ | | ____ ___ / ____ |
| | | 1 \/ 17 \/ 2 *\/ 1 - \/ 17 || | | 1 \/ 17 \/ 2 *\/ 1 - \/ 17 ||| | | |1 \/ 17 \/ 2 *\/ 1 - \/ 17 || | |1 \/ 17 \/ 2 *\/ 1 - \/ 17 ||| |1 \/ 17 \/ 2 *\/ 1 + \/ 17 | |1 \/ 17 \/ 2 *\/ 1 + \/ 17 |
-16*|I*im|atan|- - + ------ + ---------------------|| + re|atan|- - + ------ + ---------------------|||*|I*im|atan|- - ------ + ---------------------|| + re|atan|- - ------ + ---------------------|||*atan|- + ------ - ---------------------|*atan|- + ------ + ---------------------|
\ \ \ 4 4 4 // \ \ 4 4 4 /// \ \ \4 4 4 // \ \4 4 4 /// \4 4 4 / \4 4 4 /
$$- 16 \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{1}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{1 - \sqrt{17}}}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{1}{4} + \frac{\sqrt{17}}{4} + \frac{\sqrt{2} \sqrt{1 - \sqrt{17}}}{4} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(- \frac{\sqrt{17}}{4} + \frac{1}{4} + \frac{\sqrt{2} \sqrt{1 - \sqrt{17}}}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(- \frac{\sqrt{17}}{4} + \frac{1}{4} + \frac{\sqrt{2} \sqrt{1 - \sqrt{17}}}{4} \right)}\right)}\right) \operatorname{atan}{\left(\frac{1}{4} + \frac{\sqrt{2} \sqrt{1 + \sqrt{17}}}{4} + \frac{\sqrt{17}}{4} \right)} \operatorname{atan}{\left(- \frac{\sqrt{2} \sqrt{1 + \sqrt{17}}}{4} + \frac{1}{4} + \frac{\sqrt{17}}{4} \right)}$$
-16*(i*im(atan(-1/4 + sqrt(17)/4 + sqrt(2)*sqrt(1 - sqrt(17))/4)) + re(atan(-1/4 + sqrt(17)/4 + sqrt(2)*sqrt(1 - sqrt(17))/4)))*(i*im(atan(1/4 - sqrt(17)/4 + sqrt(2)*sqrt(1 - sqrt(17))/4)) + re(atan(1/4 - sqrt(17)/4 + sqrt(2)*sqrt(1 - sqrt(17))/4)))*atan(1/4 + sqrt(17)/4 - sqrt(2)*sqrt(1 + sqrt(17))/4)*atan(1/4 + sqrt(17)/4 + sqrt(2)*sqrt(1 + sqrt(17))/4)
Respuesta numérica
[src]
x1 = 330.763136108137
x2 = 66.8693532065946
x3 = -26.0286487099272
x4 = 61.935945590587
x5 = -33.6616117082788
x6 = -76.2941311673639
x7 = -2.2456851723809
x8 = -90.2102794728951
x9 = -21.0952410939197
x10 = -58.7943529369972
x11 = 68.2191308977666
x12 = 105.918242740844
x13 = 22.8870560563374
x14 = 99.6350574336645
x15 = 93.3518721264849
x16 = -46.227982322638
x17 = 55.6527602834074
x18 = 48.0197972850558
x19 = 60.586167899415
x20 = -77.6439088585359
x21 = 54.3029825922354
x22 = 4.03750013479868
x23 = -39.9447970154584
x24 = -65.0775382441768
x25 = -259.856282766744
x26 = -70.0109458601843
x27 = -82.5773164745435
x28 = -38.5950193242864
x29 = 98.2852797424925
x30 = -13.4622780955681
x31 = 24.2368337475095
x32 = -57.4445752458252
x33 = 49.3695749762278
x34 = 17.9536484403299
x35 = 74.5023162049461
x36 = -101.426872396082
x37 = 30.520019054689
x38 = 41.7366119778762
x39 = -88.8605017817231
x40 = -63.7277605530048
x41 = 16.6038707491579
x42 = 5.3872778259707
x43 = -83.9270941657155
x44 = 85.7189091281333
x45 = -27.3784264010992
x46 = 10.3206854419783
x47 = 11.6704631331503
x48 = 92.0020944353129
x49 = -3545.96219842167
x50 = -109.059835394434
x51 = 10509.5233337391
x52 = 80.7855015121257
x53 = -214.524207925315
x54 = -19.7454634027476
x55 = -32.3118340171068
x56 = -71.3607235513564
x56 = -71.3607235513564
Gráfico