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Ecuación x^2-6x+9=0
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Ecuación (x-11)^4=(x+3)^4
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Ecuación √(x^2-1)^3=2*x
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Ecuación -x/12+x=55/12
- Expresar {x} en función de y en la ecuación:
- 18*x-11*y=2
- 12*x-14*y=20
- -3*x-20*y=-13
- -16*x+14*y=-9
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Expresiones idénticas
- cos(dos *x)+cos(x)= cero , ochenta y dos / tres
- coseno de (2 multiplicar por x) más coseno de (x) es igual a 0,82 dividir por 3
- coseno de (dos multiplicar por x) más coseno de (x) es igual a cero , ochenta y dos dividir por tres
- cos(2x)+cos(x)=0,82/3
- cos2x+cosx=0,82/3
- cos(2*x)+cos(x)=O,82/3
- cos(2*x)+cos(x)=0,82 dividir por 3
-
Expresiones semejantes
- 2cos2x+cos(x/2)^2-10sinx+7/2=1/2cosx
- cos(2*x)-cos(x)=0,82/3
- sqrt(cos^2(x)+cos(x))/sin(x)+1=0
- cos^2(x)+cos(x)=sin(x)
- (cos(2x)+cos(x))/(1+sqrt(sin(x)))=0
- cos(2*x)+cosx=0,82/3
-
Expresiones con funciones
- Coseno cos
- cos(x)+sin(x)=1
- cos(2*x)=1
- cos(3*x)=0
- cos(-x)=-1
- cos(x)=-0,5
- Coseno cos
- cos(x)+sin(x)=1
- cos(2*x)=1
- cos(3*x)=0
- cos(-x)=-1
- cos(x)=-0,5
cos(2*x)+cos(x)=0,82/3 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
41
cos(2*x) + cos(x) = ----
50*3
$$\cos{\left(x \right)} + \cos{\left(2 x \right)} = \frac{41}{3 \cdot 50}$$
Solución detallada
Tenemos la ecuación
$$\cos{\left(x \right)} + \cos{\left(2 x \right)} = \frac{41}{3 \cdot 50}$$
cambiamos
$$\cos{\left(x \right)} + \cos{\left(2 x \right)} - \frac{41}{150} = 0$$
$$2 \cos^{2}{\left(x \right)} + \cos{\left(x \right)} - \frac{191}{150} = 0$$
Sustituimos
$$w = \cos{\left(x \right)}$$
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
$$a = 2$$
$$b = 1$$
$$c = - \frac{191}{150}$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = - \frac{1}{4} + \frac{\sqrt{2517}}{60}$$
$$w_{2} = - \frac{\sqrt{2517}}{60} - \frac{1}{4}$$
hacemos cambio inverso
$$\cos{\left(x \right)} = w$$
Tenemos la ecuación
$$\cos{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
O
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{1}{4} + \frac{\sqrt{2517}}{60} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{1}{4} + \frac{\sqrt{2517}}{60} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{2517}}{60} - \frac{1}{4} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{2517}}{60} - \frac{1}{4} \right)}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{4} + \frac{\sqrt{2517}}{60} \right)}$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{4} + \frac{\sqrt{2517}}{60} \right)}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{2517}}{60} - \frac{1}{4} \right)}$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{2517}}{60} - \frac{1}{4} \right)}$$
$$\cos{\left(x \right)} + \cos{\left(2 x \right)} = \frac{41}{3 \cdot 50}$$
cambiamos
$$\cos{\left(x \right)} + \cos{\left(2 x \right)} - \frac{41}{150} = 0$$
$$2 \cos^{2}{\left(x \right)} + \cos{\left(x \right)} - \frac{191}{150} = 0$$
Sustituimos
$$w = \cos{\left(x \right)}$$
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
$$a = 2$$
$$b = 1$$
$$c = - \frac{191}{150}$$
, entonces
D = b^2 - 4 * a * c =
(1)^2 - 4 * (2) * (-191/150) = 839/75
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = - \frac{1}{4} + \frac{\sqrt{2517}}{60}$$
$$w_{2} = - \frac{\sqrt{2517}}{60} - \frac{1}{4}$$
hacemos cambio inverso
$$\cos{\left(x \right)} = w$$
Tenemos la ecuación
$$\cos{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
O
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{1}{4} + \frac{\sqrt{2517}}{60} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{1}{4} + \frac{\sqrt{2517}}{60} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{2517}}{60} - \frac{1}{4} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{2517}}{60} - \frac{1}{4} \right)}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{4} + \frac{\sqrt{2517}}{60} \right)}$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{4} + \frac{\sqrt{2517}}{60} \right)}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{2517}}{60} - \frac{1}{4} \right)}$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{2517}}{60} - \frac{1}{4} \right)}$$
Respuesta rápida
[src]
/ / __________________\ \
| | ______ ___ / ______ | |
x1 = I*\- log\-15 + \/ 2517 - I*\/ 6 *\/ 143 + 5*\/ 2517 / + log(60)/
$$x_{1} = i \left(\log{\left(60 \right)} - \log{\left(-15 + \sqrt{2517} - \sqrt{6} i \sqrt{143 + 5 \sqrt{2517}} \right)}\right)$$
/ _________________________\
| / 2 |
| ______ / / ______\ |
| 1 \/ 2517 I*\/ 3600 - \15 - \/ 2517 / |
x2 = -I*log|- - + -------- + -------------------------------|
\ 4 60 60 /
$$x_{2} = - i \log{\left(- \frac{1}{4} + \frac{\sqrt{2517}}{60} + \frac{i \sqrt{3600 - \left(15 - \sqrt{2517}\right)^{2}}}{60} \right)}$$
/ 60 \
x3 = pi + I*log|--------------------------------------------|
| ___________________|
| ______ ___ / ______ |
\15 + \/ 2517 + \/ 6 *\/ -143 + 5*\/ 2517 /
$$x_{3} = \pi + i \log{\left(\frac{60}{15 + \sqrt{6} \sqrt{-143 + 5 \sqrt{2517}} + \sqrt{2517}} \right)}$$
/ 60 \
x4 = pi + I*log|--------------------------------------------|
| ___________________|
| ______ ___ / ______ |
\15 + \/ 2517 - \/ 6 *\/ -143 + 5*\/ 2517 /
$$x_{4} = \pi + i \log{\left(\frac{60}{- \sqrt{6} \sqrt{-143 + 5 \sqrt{2517}} + 15 + \sqrt{2517}} \right)}$$
x4 = pi + i*log(60/(-sqrt(6)*sqrt(-143 + 5*sqrt(2517)) + 15 + sqrt(2517)))
Suma y producto de raíces
[src]
suma
/ _________________________\
| / 2 |
/ / __________________\ \ | ______ / / ______\ |
| | ______ ___ / ______ | | | 1 \/ 2517 I*\/ 3600 - \15 - \/ 2517 / | / 60 \ / 60 \
I*\- log\-15 + \/ 2517 - I*\/ 6 *\/ 143 + 5*\/ 2517 / + log(60)/ - I*log|- - + -------- + -------------------------------| + pi + I*log|--------------------------------------------| + pi + I*log|--------------------------------------------|
\ 4 60 60 / | ___________________| | ___________________|
| ______ ___ / ______ | | ______ ___ / ______ |
\15 + \/ 2517 + \/ 6 *\/ -143 + 5*\/ 2517 / \15 + \/ 2517 - \/ 6 *\/ -143 + 5*\/ 2517 /
$$\left(\left(\pi + i \log{\left(\frac{60}{15 + \sqrt{6} \sqrt{-143 + 5 \sqrt{2517}} + \sqrt{2517}} \right)}\right) + \left(- i \log{\left(- \frac{1}{4} + \frac{\sqrt{2517}}{60} + \frac{i \sqrt{3600 - \left(15 - \sqrt{2517}\right)^{2}}}{60} \right)} + i \left(\log{\left(60 \right)} - \log{\left(-15 + \sqrt{2517} - \sqrt{6} i \sqrt{143 + 5 \sqrt{2517}} \right)}\right)\right)\right) + \left(\pi + i \log{\left(\frac{60}{- \sqrt{6} \sqrt{-143 + 5 \sqrt{2517}} + 15 + \sqrt{2517}} \right)}\right)$$
=
/ _________________________\
| / 2 |
/ / __________________\ \ | ______ / / ______\ |
| | ______ ___ / ______ | | / 60 \ / 60 \ | 1 \/ 2517 I*\/ 3600 - \15 - \/ 2517 / |
2*pi + I*\- log\-15 + \/ 2517 - I*\/ 6 *\/ 143 + 5*\/ 2517 / + log(60)/ + I*log|--------------------------------------------| + I*log|--------------------------------------------| - I*log|- - + -------- + -------------------------------|
| ___________________| | ___________________| \ 4 60 60 /
| ______ ___ / ______ | | ______ ___ / ______ |
\15 + \/ 2517 + \/ 6 *\/ -143 + 5*\/ 2517 / \15 + \/ 2517 - \/ 6 *\/ -143 + 5*\/ 2517 /
$$- i \log{\left(- \frac{1}{4} + \frac{\sqrt{2517}}{60} + \frac{i \sqrt{3600 - \left(15 - \sqrt{2517}\right)^{2}}}{60} \right)} + 2 \pi + i \log{\left(\frac{60}{15 + \sqrt{6} \sqrt{-143 + 5 \sqrt{2517}} + \sqrt{2517}} \right)} + i \left(\log{\left(60 \right)} - \log{\left(-15 + \sqrt{2517} - \sqrt{6} i \sqrt{143 + 5 \sqrt{2517}} \right)}\right) + i \log{\left(\frac{60}{- \sqrt{6} \sqrt{-143 + 5 \sqrt{2517}} + 15 + \sqrt{2517}} \right)}$$
producto
/ / _________________________\\
| | / 2 ||
/ / __________________\ \ | | ______ / / ______\ ||
| | ______ ___ / ______ | | | | 1 \/ 2517 I*\/ 3600 - \15 - \/ 2517 / || / / 60 \\ / / 60 \\
I*\- log\-15 + \/ 2517 - I*\/ 6 *\/ 143 + 5*\/ 2517 / + log(60)/*|-I*log|- - + -------- + -------------------------------||*|pi + I*log|--------------------------------------------||*|pi + I*log|--------------------------------------------||
\ \ 4 60 60 // | | ___________________|| | | ___________________||
| | ______ ___ / ______ || | | ______ ___ / ______ ||
\ \15 + \/ 2517 + \/ 6 *\/ -143 + 5*\/ 2517 // \ \15 + \/ 2517 - \/ 6 *\/ -143 + 5*\/ 2517 //
$$i \left(\log{\left(60 \right)} - \log{\left(-15 + \sqrt{2517} - \sqrt{6} i \sqrt{143 + 5 \sqrt{2517}} \right)}\right) \left(- i \log{\left(- \frac{1}{4} + \frac{\sqrt{2517}}{60} + \frac{i \sqrt{3600 - \left(15 - \sqrt{2517}\right)^{2}}}{60} \right)}\right) \left(\pi + i \log{\left(\frac{60}{15 + \sqrt{6} \sqrt{-143 + 5 \sqrt{2517}} + \sqrt{2517}} \right)}\right) \left(\pi + i \log{\left(\frac{60}{- \sqrt{6} \sqrt{-143 + 5 \sqrt{2517}} + 15 + \sqrt{2517}} \right)}\right)$$
=
/ __________________________\
| / 2 |
/ / __________________\ \ | ______ / / ______\ |
/ / 60 \\ / / 60 \\ | | ______ ___ / ______ | | | 1 \/ 2517 \/ -3600 + \15 - \/ 2517 / |
|pi + I*log|--------------------------------------------||*|pi + I*log|--------------------------------------------||*\- log\-15 + \/ 2517 - I*\/ 6 *\/ 143 + 5*\/ 2517 / + log(60)/*log|- - + -------- + ------------------------------|
| | ___________________|| | | ___________________|| \ 4 60 60 /
| | ______ ___ / ______ || | | ______ ___ / ______ ||
\ \15 + \/ 2517 + \/ 6 *\/ -143 + 5*\/ 2517 // \ \15 + \/ 2517 - \/ 6 *\/ -143 + 5*\/ 2517 //
$$\left(\pi + i \log{\left(\frac{60}{15 + \sqrt{6} \sqrt{-143 + 5 \sqrt{2517}} + \sqrt{2517}} \right)}\right) \left(\pi + i \log{\left(\frac{60}{- \sqrt{6} \sqrt{-143 + 5 \sqrt{2517}} + 15 + \sqrt{2517}} \right)}\right) \left(\log{\left(60 \right)} - \log{\left(-15 + \sqrt{2517} - \sqrt{6} i \sqrt{143 + 5 \sqrt{2517}} \right)}\right) \log{\left(- \frac{1}{4} + \frac{\sqrt{2517}}{60} + \frac{\sqrt{-3600 + \left(15 - \sqrt{2517}\right)^{2}}}{60} \right)}$$
(pi + i*log(60/(15 + sqrt(2517) + sqrt(6)*sqrt(-143 + 5*sqrt(2517)))))*(pi + i*log(60/(15 + sqrt(2517) - sqrt(6)*sqrt(-143 + 5*sqrt(2517)))))*(-log(-15 + sqrt(2517) - i*sqrt(6)*sqrt(143 + 5*sqrt(2517))) + log(60))*log(-1/4 + sqrt(2517)/60 + sqrt(-3600 + (15 - sqrt(2517))^2)/60)
Respuesta numérica
[src]
x1 = 70.0595213312407
x2 = -57.4931507168816
x3 = 32.3604094881632
x4 = -30.4714435836327
x5 = -74.4537407338898
x6 = 63.7763360240611
x7 = 82.6258919455999
x8 = 99.5864819626081
x9 = -76.3427066384203
x10 = -7.22766825944487
x11 = -24.1882582764531
x12 = 30.4714435836327
x13 = 269.232485256457
x14 = 11.6218876620939
x15 = 13.5108535666245
x16 = -43.0378141979918
x17 = -26.0772241809836
x18 = 93.3032966554285
x19 = -82.6258919455999
x20 = -11.6218876620939
x21 = 76.3427066384203
x22 = 26.0772241809836
x23 = -51.209965409702
x24 = 95.1922625599591
x25 = 24.1882582764531
x26 = -93.3032966554285
x27 = 38.6435947953428
x28 = -5.3387023549143
x29 = -13.5108535666245
x30 = 55.604184812351
x31 = 80.7369260410693
x32 = -55.604184812351
x33 = -32.3604094881632
x34 = 61.8873701195306
x35 = -17.9050729692735
x36 = 51.209965409702
x37 = -68.1705554267102
x38 = -99.5864819626081
x39 = 57.4931507168816
x40 = 36.7546288908122
x41 = -70.0595213312407
x42 = 17.9050729692735
x43 = -19.794038873804
x44 = -61.8873701195306
x45 = -49.3209995051714
x46 = 68.1705554267102
x47 = 74.4537407338898
x48 = -38.6435947953428
x49 = -95.1922625599591
x50 = 19.794038873804
x51 = 131.002408498506
x52 = -421.917898533298
x53 = 49.3209995051714
x54 = -63.7763360240611
x55 = -0.944482952265282
x56 = -139.174559710216
x56 = -139.174559710216