Sr Examen
Otras calculadoras
-
¿Cómo usar?
- La ecuación:
-
Ecuación x^2+5x=36
-
Ecuación 3*x-6=x+4
-
Ecuación 9^x-8*3^x-9=0
-
Ecuación x^3+4=0
- Expresar {x} en función de y en la ecuación:
- -2*x-19*y=3
- 6*x+17*y=-11
- -11*x-7*y=-2
- -19*x+14*y=20
-
Expresiones idénticas
- √(tres +4cos dos x)=√2*cosx
- √(3 más 4 coseno de 2x) es igual a √2 multiplicar por coseno de x
- √(tres más 4 coseno de dos x) es igual a √2 multiplicar por coseno de x
- √(3+4cos2x)=√2cosx
- √3+4cos2x=√2cosx
-
Expresiones semejantes
- √(2)cosx-1=0
- 2sin(x)^3-sin(x)+2√2=2√2cos(x)
- √(3+4cos2x)=√2cosx
- √(3-4cos2x)=√2*cosx
- 2sinx^3+2√2cosx^2+sinx=2√2
-
Expresiones con funciones
- cosx
- cosx-2|cosx|=y
√(3+4cos2x)=√2*cosx la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
________________ ___ \/ 3 + 4*cos(2*x) = \/ 2 *cos(x)
$$\sqrt{4 \cos{\left(2 x \right)} + 3} = \sqrt{2} \cos{\left(x \right)}$$
Solución detallada
Tenemos la ecuación
$$\sqrt{4 \cos{\left(2 x \right)} + 3} = \sqrt{2} \cos{\left(x \right)}$$
cambiamos
$$\sqrt{8 \cos^{2}{\left(x \right)} + 3} - \sqrt{2} \cos{\left(x \right)} - 1 = 0$$
$$\sqrt{8 \cos^{2}{\left(x \right)} + 3} - \sqrt{2} \cos{\left(x \right)} - 1 = 0$$
Sustituimos
$$w = \cos{\left(x \right)}$$
$$\sqrt{8 w^{2} + 3} = \sqrt{2} w + 1$$
Elevemos las dos partes de la ecuación a la potencia 2
$$8 w^{2} + 3 = \left(\sqrt{2} w + 1\right)^{2}$$
$$8 w^{2} + 3 = 2 w^{2} + 2 \sqrt{2} w + 1$$
Transpongamos la parte derecha de la ecuación miembro izquierdo de la ecuación con el signo negativo
$$6 w^{2} - 2 \sqrt{2} w + 2 = 0$$
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
$$a = 6$$
$$b = - 2 \sqrt{2}$$
$$c = 2$$
, entonces
Como D < 0 la ecuación
no tiene raíces reales,
pero hay raíces complejas.
o
$$w_{1} = \frac{\sqrt{2}}{6} + \frac{\sqrt{10} i}{6}$$
$$w_{2} = \frac{\sqrt{2}}{6} - \frac{\sqrt{10} i}{6}$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{\sqrt{2}}{6} + \frac{\sqrt{10} i}{6} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{\sqrt{2}}{6} + \frac{\sqrt{10} i}{6} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(\frac{\sqrt{2}}{6} - \frac{\sqrt{10} i}{6} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(\frac{\sqrt{2}}{6} - \frac{\sqrt{10} i}{6} \right)}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{2}}{6} + \frac{\sqrt{10} i}{6} \right)}$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{2}}{6} + \frac{\sqrt{10} i}{6} \right)}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{2}}{6} - \frac{\sqrt{10} i}{6} \right)}$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{2}}{6} - \frac{\sqrt{10} i}{6} \right)}$$
$$\sqrt{4 \cos{\left(2 x \right)} + 3} = \sqrt{2} \cos{\left(x \right)}$$
cambiamos
$$\sqrt{8 \cos^{2}{\left(x \right)} + 3} - \sqrt{2} \cos{\left(x \right)} - 1 = 0$$
$$\sqrt{8 \cos^{2}{\left(x \right)} + 3} - \sqrt{2} \cos{\left(x \right)} - 1 = 0$$
Sustituimos
$$w = \cos{\left(x \right)}$$
$$\sqrt{8 w^{2} + 3} = \sqrt{2} w + 1$$
Elevemos las dos partes de la ecuación a la potencia 2
$$8 w^{2} + 3 = \left(\sqrt{2} w + 1\right)^{2}$$
$$8 w^{2} + 3 = 2 w^{2} + 2 \sqrt{2} w + 1$$
Transpongamos la parte derecha de la ecuación miembro izquierdo de la ecuación con el signo negativo
$$6 w^{2} - 2 \sqrt{2} w + 2 = 0$$
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
$$a = 6$$
$$b = - 2 \sqrt{2}$$
$$c = 2$$
, entonces
D = b^2 - 4 * a * c =
(-2*sqrt(2))^2 - 4 * (6) * (2) = -40
Como D < 0 la ecuación
no tiene raíces reales,
pero hay raíces complejas.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = \frac{\sqrt{2}}{6} + \frac{\sqrt{10} i}{6}$$
$$w_{2} = \frac{\sqrt{2}}{6} - \frac{\sqrt{10} i}{6}$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{\sqrt{2}}{6} + \frac{\sqrt{10} i}{6} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{\sqrt{2}}{6} + \frac{\sqrt{10} i}{6} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(\frac{\sqrt{2}}{6} - \frac{\sqrt{10} i}{6} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(\frac{\sqrt{2}}{6} - \frac{\sqrt{10} i}{6} \right)}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{2}}{6} + \frac{\sqrt{10} i}{6} \right)}$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{2}}{6} + \frac{\sqrt{10} i}{6} \right)}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{2}}{6} - \frac{\sqrt{10} i}{6} \right)}$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{2}}{6} - \frac{\sqrt{10} i}{6} \right)}$$
Respuesta rápida
[src]
/ / / ___\\\
| | |\/ 5 ||| / _______________________________________\
| |atan|-----||| | / / / ___\\ / / ___\\ |
| | \ 2 /|| | / | |\/ 5 || | |\/ 5 || |
|cos|-----------|| | / |atan|-----|| |atan|-----|| |
| \ 2 /| | / 2| \ 2 /| 2| \ 2 /| |
x1 = - atan|----------------| - I*log| / cos |-----------| + sin |-----------| |
| / / ___\\| \\/ \ 2 / \ 2 / /
| | |\/ 5 |||
| |atan|-----|||
| | \ 2 /||
|sin|-----------||
\ \ 2 //
$$x_{1} = - \operatorname{atan}{\left(\frac{\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)}}{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)} + \cos^{2}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)}} \right)}$$
/ / / ___\\\
/ _______________________________________\ | | |\/ 5 |||
| / / / ___\\ / / ___\\ | | |atan|-----|||
| / | |\/ 5 || | |\/ 5 || | | | \ 2 /||
| / |atan|-----|| |atan|-----|| | |cos|-----------||
| / 2| \ 2 /| 2| \ 2 /| | | \ 2 /|
x2 = - I*log| / cos |-----------| + sin |-----------| | + atan|----------------|
\\/ \ 2 / \ 2 / / | / / ___\\|
| | |\/ 5 |||
| |atan|-----|||
| | \ 2 /||
|sin|-----------||
\ \ 2 //
$$x_{2} = - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)} + \cos^{2}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)}} \right)} + \operatorname{atan}{\left(\frac{\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)}}{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)}} \right)}$$
x2 = -i*log(sqrt(sin(atan(sqrt(5)/2)/2)^2 + cos(atan(sqrt(5)/2)/2)^2)) + atan(cos(atan(sqrt(5)/2)/2)/sin(atan(sqrt(5)/2)/2))
Suma y producto de raíces
[src]
suma
/ / / ___\\\ / / / ___\\\
| | |\/ 5 ||| / _______________________________________\ / _______________________________________\ | | |\/ 5 |||
| |atan|-----||| | / / / ___\\ / / ___\\ | | / / / ___\\ / / ___\\ | | |atan|-----|||
| | \ 2 /|| | / | |\/ 5 || | |\/ 5 || | | / | |\/ 5 || | |\/ 5 || | | | \ 2 /||
|cos|-----------|| | / |atan|-----|| |atan|-----|| | | / |atan|-----|| |atan|-----|| | |cos|-----------||
| \ 2 /| | / 2| \ 2 /| 2| \ 2 /| | | / 2| \ 2 /| 2| \ 2 /| | | \ 2 /|
- atan|----------------| - I*log| / cos |-----------| + sin |-----------| | + - I*log| / cos |-----------| + sin |-----------| | + atan|----------------|
| / / ___\\| \\/ \ 2 / \ 2 / / \\/ \ 2 / \ 2 / / | / / ___\\|
| | |\/ 5 ||| | | |\/ 5 |||
| |atan|-----||| | |atan|-----|||
| | \ 2 /|| | | \ 2 /||
|sin|-----------|| |sin|-----------||
\ \ 2 // \ \ 2 //
$$\left(- \operatorname{atan}{\left(\frac{\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)}}{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)} + \cos^{2}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)}} \right)}\right) + \left(- i \log{\left(\sqrt{\sin^{2}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)} + \cos^{2}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)}} \right)} + \operatorname{atan}{\left(\frac{\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)}}{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)}} \right)}\right)$$
=
/ _______________________________________\
| / / / ___\\ / / ___\\ |
| / | |\/ 5 || | |\/ 5 || |
| / |atan|-----|| |atan|-----|| |
| / 2| \ 2 /| 2| \ 2 /| |
-2*I*log| / cos |-----------| + sin |-----------| |
\\/ \ 2 / \ 2 / /
$$- 2 i \log{\left(\sqrt{\sin^{2}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)} + \cos^{2}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)}} \right)}$$
producto
/ / / / ___\\\ \ / / / / ___\\\\ | | | |\/ 5 ||| / _______________________________________\| | / _______________________________________\ | | |\/ 5 |||| | | |atan|-----||| | / / / ___\\ / / ___\\ || | | / / / ___\\ / / ___\\ | | |atan|-----|||| | | | \ 2 /|| | / | |\/ 5 || | |\/ 5 || || | | / | |\/ 5 || | |\/ 5 || | | | \ 2 /||| | |cos|-----------|| | / |atan|-----|| |atan|-----|| || | | / |atan|-----|| |atan|-----|| | |cos|-----------||| | | \ 2 /| | / 2| \ 2 /| 2| \ 2 /| || | | / 2| \ 2 /| 2| \ 2 /| | | \ 2 /|| |- atan|----------------| - I*log| / cos |-----------| + sin |-----------| ||*|- I*log| / cos |-----------| + sin |-----------| | + atan|----------------|| | | / / ___\\| \\/ \ 2 / \ 2 / /| | \\/ \ 2 / \ 2 / / | / / ___\\|| | | | |\/ 5 ||| | | | | |\/ 5 |||| | | |atan|-----||| | | | |atan|-----|||| | | | \ 2 /|| | | | | \ 2 /||| | |sin|-----------|| | | |sin|-----------||| \ \ \ 2 // / \ \ \ 2 ///
$$\left(- i \log{\left(\sqrt{\sin^{2}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)} + \cos^{2}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)}} \right)} + \operatorname{atan}{\left(\frac{\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)}}{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)}} \right)}\right) \left(- \operatorname{atan}{\left(\frac{\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)}}{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)}} \right)} - i \log{\left(\sqrt{\sin^{2}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)} + \cos^{2}{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}}{2} \right)}} \right)}\right)$$
=
2
/ / ___\\
| |\/ 5 ||
-|pi - atan|-----||
\ \ 2 //
---------------------
4
$$- \frac{\left(\pi - \operatorname{atan}{\left(\frac{\sqrt{5}}{2} \right)}\right)^{2}}{4}$$
-(pi - atan(sqrt(5)/2))^2/4
Respuesta numérica
[src]
x1 = -70.2653003704864
x2 = 13.7166326058701
x3 = -38.8493738345884
x4 = -19.9998179130497
x5 = 74.2479616946441
x6 = -11.4161086228482
x7 = 61.6815910802849
x8 = 11.4161086228482
x9 = -26.2830032202293
x10 = 55.3984057731053
x11 = -57.6989297561272
x12 = -61.6815910802849
x13 = -17.6992939300278
x14 = 38.8493738345884
x15 = -55.3984057731053
x16 = 82.8316709848456
x17 = 30.265664544387
x18 = 19.9998179130497
x19 = 70.2653003704864
x20 = -63.9821150633068
x21 = -99.3807029233624
x22 = 23.9824792372074
x23 = 26.2830032202293
x24 = -5.13292331566865
x25 = -76.548485677666
x26 = 7.43344729869052
x27 = 32.5661885274089
x28 = -679.734275166906
x29 = -51.4157444489476
x30 = -23.9824792372074
x31 = -7.43344729869052
x32 = -67.9647763874645
x33 = 99.3807029233624
x34 = 67.9647763874645
x35 = -32.5661885274089
x36 = 1.15026199151093
x37 = -13.7166326058701
x38 = 76.548485677666
x39 = 63.9821150633068
x40 = 80.5311470018237
x41 = 17.6992939300278
x42 = -30.265664544387
x43 = -95.3980415992047
x44 = 93.0975176161829
x44 = 93.0975176161829