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Expresiones idénticas
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- cos(2*x)+sin(x)+1=O
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Expresiones semejantes
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Expresiones con funciones
- Coseno cos
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- cos(3*x)=0
- cos(-x)=-1
- cos(x)=-0,5
- Seno sin
- sin(x)=-1/5
- sin(3*x-pi/6)=1/2
- sin(x)=-3/2
- sin(x)^3+cos(x)^3=1
- sin(pi*x/3)=1/2
cos(2*x)+sin(x)+1=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
cos(2*x) + sin(x) + 1 = 0
$$\left(\sin{\left(x \right)} + \cos{\left(2 x \right)}\right) + 1 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(\sin{\left(x \right)} + \cos{\left(2 x \right)}\right) + 1 = 0$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} - \frac{\sqrt{17}}{4} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} - \frac{\sqrt{17}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} + \frac{\sqrt{17}}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} + \frac{\sqrt{17}}{4} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} - \frac{\sqrt{17}}{4} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} - \frac{\sqrt{17}}{4} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{1}{4} + \frac{\sqrt{17}}{4} \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{1}{4} + \frac{\sqrt{17}}{4} \right)}$$
$$\left(\sin{\left(x \right)} + \cos{\left(2 x \right)}\right) + 1 = 0$$
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
[src]
/ _____________ \
| ___ / ____ / ____\|
| \/ 2 *\/ -1 + \/ 17 I*\1 - \/ 17 /|
x1 = -I*log|- ---------------------- + --------------|
\ 4 4 /
$$x_{1} = - i \log{\left(- \frac{\sqrt{2} \sqrt{-1 + \sqrt{17}}}{4} + \frac{i \left(1 - \sqrt{17}\right)}{4} \right)}$$
/ _____________\
| / ____\ ___ / ____ |
|I*\1 - \/ 17 / \/ 2 *\/ -1 + \/ 17 |
x2 = -I*log|-------------- + ----------------------|
\ 4 4 /
$$x_{2} = - i \log{\left(\frac{\sqrt{2} \sqrt{-1 + \sqrt{17}}}{4} + \frac{i \left(1 - \sqrt{17}\right)}{4} \right)}$$
/ 2\
|/ ____________\ |
|| ____ ___ / ____ | |
|\1 + \/ 17 + \/ 2 *\/ 1 + \/ 17 / |
I*log|-------------------------------------|
pi \ 16 /
x3 = -- - --------------------------------------------
2 2
$$x_{3} = \frac{\pi}{2} - \frac{i \log{\left(\frac{\left(1 + \sqrt{2} \sqrt{1 + \sqrt{17}} + \sqrt{17}\right)^{2}}{16} \right)}}{2}$$
/ 2\
|/ ____________\ |
|| ____ ___ / ____ | |
|\1 + \/ 17 - \/ 2 *\/ 1 + \/ 17 / |
I*log|-------------------------------------|
pi \ 16 /
x4 = -- - --------------------------------------------
2 2
$$x_{4} = \frac{\pi}{2} - \frac{i \log{\left(\frac{\left(- \sqrt{2} \sqrt{1 + \sqrt{17}} + 1 + \sqrt{17}\right)^{2}}{16} \right)}}{2}$$
x4 = pi/2 - i*log((-sqrt(2)*sqrt(1 + sqrt(17)) + 1 + sqrt(17))^2/16)/2
Suma y producto de raíces
[src]
suma
/ 2\ / 2\
|/ ____________\ | |/ ____________\ |
|| ____ ___ / ____ | | || ____ ___ / ____ | |
/ _____________ \ / _____________\ |\1 + \/ 17 + \/ 2 *\/ 1 + \/ 17 / | |\1 + \/ 17 - \/ 2 *\/ 1 + \/ 17 / |
| ___ / ____ / ____\| | / ____\ ___ / ____ | I*log|-------------------------------------| I*log|-------------------------------------|
| \/ 2 *\/ -1 + \/ 17 I*\1 - \/ 17 /| |I*\1 - \/ 17 / \/ 2 *\/ -1 + \/ 17 | pi \ 16 / pi \ 16 /
- I*log|- ---------------------- + --------------| - I*log|-------------- + ----------------------| + -- - -------------------------------------------- + -- - --------------------------------------------
\ 4 4 / \ 4 4 / 2 2 2 2
$$\left(\left(- i \log{\left(- \frac{\sqrt{2} \sqrt{-1 + \sqrt{17}}}{4} + \frac{i \left(1 - \sqrt{17}\right)}{4} \right)} - i \log{\left(\frac{\sqrt{2} \sqrt{-1 + \sqrt{17}}}{4} + \frac{i \left(1 - \sqrt{17}\right)}{4} \right)}\right) + \left(\frac{\pi}{2} - \frac{i \log{\left(\frac{\left(1 + \sqrt{2} \sqrt{1 + \sqrt{17}} + \sqrt{17}\right)^{2}}{16} \right)}}{2}\right)\right) + \left(\frac{\pi}{2} - \frac{i \log{\left(\frac{\left(- \sqrt{2} \sqrt{1 + \sqrt{17}} + 1 + \sqrt{17}\right)^{2}}{16} \right)}}{2}\right)$$
=
/ 2\ / 2\
|/ ____________\ | |/ ____________\ |
|| ____ ___ / ____ | | || ____ ___ / ____ | |
/ _____________ \ / _____________\ |\1 + \/ 17 + \/ 2 *\/ 1 + \/ 17 / | |\1 + \/ 17 - \/ 2 *\/ 1 + \/ 17 / |
| ___ / ____ / ____\| | / ____\ ___ / ____ | I*log|-------------------------------------| I*log|-------------------------------------|
| \/ 2 *\/ -1 + \/ 17 I*\1 - \/ 17 /| |I*\1 - \/ 17 / \/ 2 *\/ -1 + \/ 17 | \ 16 / \ 16 /
pi - I*log|- ---------------------- + --------------| - I*log|-------------- + ----------------------| - -------------------------------------------- - --------------------------------------------
\ 4 4 / \ 4 4 / 2 2
$$- i \log{\left(- \frac{\sqrt{2} \sqrt{-1 + \sqrt{17}}}{4} + \frac{i \left(1 - \sqrt{17}\right)}{4} \right)} - i \log{\left(\frac{\sqrt{2} \sqrt{-1 + \sqrt{17}}}{4} + \frac{i \left(1 - \sqrt{17}\right)}{4} \right)} + \pi - \frac{i \log{\left(\frac{\left(1 + \sqrt{2} \sqrt{1 + \sqrt{17}} + \sqrt{17}\right)^{2}}{16} \right)}}{2} - \frac{i \log{\left(\frac{\left(- \sqrt{2} \sqrt{1 + \sqrt{17}} + 1 + \sqrt{17}\right)^{2}}{16} \right)}}{2}$$
producto
/ / 2\\ / / 2\\
| |/ ____________\ || | |/ ____________\ ||
| || ____ ___ / ____ | || | || ____ ___ / ____ | ||
/ _____________ \ / / _____________\\ | |\1 + \/ 17 + \/ 2 *\/ 1 + \/ 17 / || | |\1 + \/ 17 - \/ 2 *\/ 1 + \/ 17 / ||
| ___ / ____ / ____\| | | / ____\ ___ / ____ || | I*log|-------------------------------------|| | I*log|-------------------------------------||
| \/ 2 *\/ -1 + \/ 17 I*\1 - \/ 17 /| | |I*\1 - \/ 17 / \/ 2 *\/ -1 + \/ 17 || |pi \ 16 /| |pi \ 16 /|
-I*log|- ---------------------- + --------------|*|-I*log|-------------- + ----------------------||*|-- - --------------------------------------------|*|-- - --------------------------------------------|
\ 4 4 / \ \ 4 4 // \2 2 / \2 2 /
$$- i \log{\left(- \frac{\sqrt{2} \sqrt{-1 + \sqrt{17}}}{4} + \frac{i \left(1 - \sqrt{17}\right)}{4} \right)} \left(- i \log{\left(\frac{\sqrt{2} \sqrt{-1 + \sqrt{17}}}{4} + \frac{i \left(1 - \sqrt{17}\right)}{4} \right)}\right) \left(\frac{\pi}{2} - \frac{i \log{\left(\frac{\left(1 + \sqrt{2} \sqrt{1 + \sqrt{17}} + \sqrt{17}\right)^{2}}{16} \right)}}{2}\right) \left(\frac{\pi}{2} - \frac{i \log{\left(\frac{\left(- \sqrt{2} \sqrt{1 + \sqrt{17}} + 1 + \sqrt{17}\right)^{2}}{16} \right)}}{2}\right)$$
=
/ / 2\\ / / 2\\
| |/ ____________\ || | |/ ____________\ || / _____________ \ / _____________\
| || ____ ___ / ____ | || | || ____ ___ / ____ | || | ___ / ____ / ____\| | / ____\ ___ / ____ |
| |\1 + \/ 17 + \/ 2 *\/ 1 + \/ 17 / || | |\1 + \/ 17 - \/ 2 *\/ 1 + \/ 17 / || | \/ 2 *\/ -1 + \/ 17 I*\1 - \/ 17 /| |I*\1 - \/ 17 / \/ 2 *\/ -1 + \/ 17 |
-|pi - I*log|-------------------------------------||*|pi - I*log|-------------------------------------||*log|- ---------------------- + --------------|*log|-------------- + ----------------------|
\ \ 16 // \ \ 16 // \ 4 4 / \ 4 4 /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
4
$$- \frac{\left(\pi - i \log{\left(\frac{\left(1 + \sqrt{2} \sqrt{1 + \sqrt{17}} + \sqrt{17}\right)^{2}}{16} \right)}\right) \left(\pi - i \log{\left(\frac{\left(- \sqrt{2} \sqrt{1 + \sqrt{17}} + 1 + \sqrt{17}\right)^{2}}{16} \right)}\right) \log{\left(- \frac{\sqrt{2} \sqrt{-1 + \sqrt{17}}}{4} + \frac{i \left(1 - \sqrt{17}\right)}{4} \right)} \log{\left(\frac{\sqrt{2} \sqrt{-1 + \sqrt{17}}}{4} + \frac{i \left(1 - \sqrt{17}\right)}{4} \right)}}{4}$$
-(pi - i*log((1 + sqrt(17) + sqrt(2)*sqrt(1 + sqrt(17)))^2/16))*(pi - i*log((1 + sqrt(17) - sqrt(2)*sqrt(1 + sqrt(17)))^2/16))*log(-sqrt(2)*sqrt(-1 + sqrt(17))/4 + i*(1 - sqrt(17))/4)*log(i*(1 - sqrt(17))/4 + sqrt(2)*sqrt(-1 + sqrt(17))/4)/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 = -1918.61720386215
x41 = -63.7277605530048
x42 = 16.6038707491579
x43 = 274.214468343521
x44 = 5.3872778259707
x45 = -83.9270941657155
x46 = 85.7189091281333
x47 = -27.3784264010992
x48 = 10.3206854419783
x49 = 11.6704631331503
x50 = 92.0020944353129
x51 = -3545.96219842167
x52 = -109.059835394434
x53 = 80.7855015121257
x54 = -214.524207925315
x55 = -19.7454634027476
x56 = -32.3118340171068
x57 = -71.3607235513564
x57 = -71.3607235513564
Gráfico