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Expresiones idénticas
- cosx+4cos2x= cero
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cosx+4cos2x=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
cos(x) + 4*cos(2*x) = 0
$$\cos{\left(x \right)} + 4 \cos{\left(2 x \right)} = 0$$
Solución detallada
Tenemos la ecuación
$$\cos{\left(x \right)} + 4 \cos{\left(2 x \right)} = 0$$
cambiamos
$$8 \cos^{2}{\left(x \right)} + \cos{\left(x \right)} - 1 = 0$$
$$8 \cos^{2}{\left(x \right)} + \cos{\left(x \right)} - 1 = 0$$
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
$$a = 8$$
$$b = 1$$
$$c = -1$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = - \frac{1}{16} + \frac{\sqrt{33}}{16}$$
$$w_{2} = - \frac{\sqrt{33}}{16} - \frac{1}{16}$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{1}{16} + \frac{\sqrt{33}}{16} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{1}{16} + \frac{\sqrt{33}}{16} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{33}}{16} - \frac{1}{16} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{33}}{16} - \frac{1}{16} \right)}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{16} + \frac{\sqrt{33}}{16} \right)}$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{16} + \frac{\sqrt{33}}{16} \right)}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{33}}{16} - \frac{1}{16} \right)}$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{33}}{16} - \frac{1}{16} \right)}$$
$$\cos{\left(x \right)} + 4 \cos{\left(2 x \right)} = 0$$
cambiamos
$$8 \cos^{2}{\left(x \right)} + \cos{\left(x \right)} - 1 = 0$$
$$8 \cos^{2}{\left(x \right)} + \cos{\left(x \right)} - 1 = 0$$
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
$$a = 8$$
$$b = 1$$
$$c = -1$$
, entonces
D = b^2 - 4 * a * c =
(1)^2 - 4 * (8) * (-1) = 33
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = - \frac{1}{16} + \frac{\sqrt{33}}{16}$$
$$w_{2} = - \frac{\sqrt{33}}{16} - \frac{1}{16}$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{1}{16} + \frac{\sqrt{33}}{16} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(- \frac{1}{16} + \frac{\sqrt{33}}{16} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{33}}{16} - \frac{1}{16} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{33}}{16} - \frac{1}{16} \right)}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{16} + \frac{\sqrt{33}}{16} \right)}$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{16} + \frac{\sqrt{33}}{16} \right)}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{33}}{16} - \frac{1}{16} \right)}$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{33}}{16} - \frac{1}{16} \right)}$$
Respuesta rápida
[src]
/ ______________\
| _____ ___ / _____ |
| 1 \/ 129 I*\/ 2 *\/ 63 + \/ 129 |
x1 = -I*log|- -- + ------- - -------------------------|
\ 16 16 16 /
$$x_{1} = - i \log{\left(- \frac{1}{16} + \frac{\sqrt{129}}{16} - \frac{\sqrt{2} i \sqrt{\sqrt{129} + 63}}{16} \right)}$$
/ ______________\
| _____ ___ / _____ |
| 1 \/ 129 I*\/ 2 *\/ 63 + \/ 129 |
x2 = -I*log|- -- + ------- + -------------------------|
\ 16 16 16 /
$$x_{2} = - i \log{\left(- \frac{1}{16} + \frac{\sqrt{129}}{16} + \frac{\sqrt{2} i \sqrt{\sqrt{129} + 63}}{16} \right)}$$
/ ______________\
| _____ ___ / _____ |
| 1 \/ 129 I*\/ 2 *\/ 63 - \/ 129 |
x3 = -I*log|- -- - ------- - -------------------------|
\ 16 16 16 /
$$x_{3} = - i \log{\left(- \frac{\sqrt{129}}{16} - \frac{1}{16} - \frac{\sqrt{2} i \sqrt{63 - \sqrt{129}}}{16} \right)}$$
/ ______________\
| _____ ___ / _____ |
| 1 \/ 129 I*\/ 2 *\/ 63 - \/ 129 |
x4 = -I*log|- -- - ------- + -------------------------|
\ 16 16 16 /
$$x_{4} = - i \log{\left(- \frac{\sqrt{129}}{16} - \frac{1}{16} + \frac{\sqrt{2} i \sqrt{63 - \sqrt{129}}}{16} \right)}$$
x4 = -i*log(-sqrt(129)/16 - 1/16 + sqrt(2)*i*sqrt(63 - sqrt(129))/16)
Suma y producto de raíces
[src]
suma
/ ______________\ / ______________\ / ______________\ / ______________\
| _____ ___ / _____ | | _____ ___ / _____ | | _____ ___ / _____ | | _____ ___ / _____ |
| 1 \/ 129 I*\/ 2 *\/ 63 + \/ 129 | | 1 \/ 129 I*\/ 2 *\/ 63 + \/ 129 | | 1 \/ 129 I*\/ 2 *\/ 63 - \/ 129 | | 1 \/ 129 I*\/ 2 *\/ 63 - \/ 129 |
- I*log|- -- + ------- - -------------------------| - I*log|- -- + ------- + -------------------------| - I*log|- -- - ------- - -------------------------| - I*log|- -- - ------- + -------------------------|
\ 16 16 16 / \ 16 16 16 / \ 16 16 16 / \ 16 16 16 /
$$\left(- i \log{\left(- \frac{\sqrt{129}}{16} - \frac{1}{16} - \frac{\sqrt{2} i \sqrt{63 - \sqrt{129}}}{16} \right)} + \left(- i \log{\left(- \frac{1}{16} + \frac{\sqrt{129}}{16} - \frac{\sqrt{2} i \sqrt{\sqrt{129} + 63}}{16} \right)} - i \log{\left(- \frac{1}{16} + \frac{\sqrt{129}}{16} + \frac{\sqrt{2} i \sqrt{\sqrt{129} + 63}}{16} \right)}\right)\right) - i \log{\left(- \frac{\sqrt{129}}{16} - \frac{1}{16} + \frac{\sqrt{2} i \sqrt{63 - \sqrt{129}}}{16} \right)}$$
=
/ ______________\ / ______________\ / ______________\ / ______________\
| _____ ___ / _____ | | _____ ___ / _____ | | _____ ___ / _____ | | _____ ___ / _____ |
| 1 \/ 129 I*\/ 2 *\/ 63 - \/ 129 | | 1 \/ 129 I*\/ 2 *\/ 63 - \/ 129 | | 1 \/ 129 I*\/ 2 *\/ 63 + \/ 129 | | 1 \/ 129 I*\/ 2 *\/ 63 + \/ 129 |
- I*log|- -- - ------- - -------------------------| - I*log|- -- - ------- + -------------------------| - I*log|- -- + ------- - -------------------------| - I*log|- -- + ------- + -------------------------|
\ 16 16 16 / \ 16 16 16 / \ 16 16 16 / \ 16 16 16 /
$$- i \log{\left(- \frac{\sqrt{129}}{16} - \frac{1}{16} - \frac{\sqrt{2} i \sqrt{63 - \sqrt{129}}}{16} \right)} - i \log{\left(- \frac{1}{16} + \frac{\sqrt{129}}{16} - \frac{\sqrt{2} i \sqrt{\sqrt{129} + 63}}{16} \right)} - i \log{\left(- \frac{1}{16} + \frac{\sqrt{129}}{16} + \frac{\sqrt{2} i \sqrt{\sqrt{129} + 63}}{16} \right)} - i \log{\left(- \frac{\sqrt{129}}{16} - \frac{1}{16} + \frac{\sqrt{2} i \sqrt{63 - \sqrt{129}}}{16} \right)}$$
producto
/ ______________\ / / ______________\\ / / ______________\\ / / ______________\\
| _____ ___ / _____ | | | _____ ___ / _____ || | | _____ ___ / _____ || | | _____ ___ / _____ ||
| 1 \/ 129 I*\/ 2 *\/ 63 + \/ 129 | | | 1 \/ 129 I*\/ 2 *\/ 63 + \/ 129 || | | 1 \/ 129 I*\/ 2 *\/ 63 - \/ 129 || | | 1 \/ 129 I*\/ 2 *\/ 63 - \/ 129 ||
-I*log|- -- + ------- - -------------------------|*|-I*log|- -- + ------- + -------------------------||*|-I*log|- -- - ------- - -------------------------||*|-I*log|- -- - ------- + -------------------------||
\ 16 16 16 / \ \ 16 16 16 // \ \ 16 16 16 // \ \ 16 16 16 //
$$- i \log{\left(- \frac{\sqrt{129}}{16} - \frac{1}{16} + \frac{\sqrt{2} i \sqrt{63 - \sqrt{129}}}{16} \right)} - i \log{\left(- \frac{\sqrt{129}}{16} - \frac{1}{16} - \frac{\sqrt{2} i \sqrt{63 - \sqrt{129}}}{16} \right)} - i \log{\left(- \frac{1}{16} + \frac{\sqrt{129}}{16} - \frac{\sqrt{2} i \sqrt{\sqrt{129} + 63}}{16} \right)} \left(- i \log{\left(- \frac{1}{16} + \frac{\sqrt{129}}{16} + \frac{\sqrt{2} i \sqrt{\sqrt{129} + 63}}{16} \right)}\right)$$
=
/ ______________\ / ______________\ / _______________\ / ______________\ | _____ ___ / _____ | | _____ ___ / _____ | | _____ ___ / _____ | | _____ ___ / _____ | | 1 \/ 129 I*\/ 2 *\/ 63 - \/ 129 | | 1 \/ 129 I*\/ 2 *\/ 63 - \/ 129 | | 1 \/ 129 \/ 2 *\/ -63 - \/ 129 | | 1 \/ 129 I*\/ 2 *\/ 63 + \/ 129 | log|- -- - ------- - -------------------------|*log|- -- - ------- + -------------------------|*log|- -- + ------- + ------------------------|*log|- -- + ------- - -------------------------| \ 16 16 16 / \ 16 16 16 / \ 16 16 16 / \ 16 16 16 /
$$\log{\left(- \frac{1}{16} + \frac{\sqrt{129}}{16} + \frac{\sqrt{2} \sqrt{-63 - \sqrt{129}}}{16} \right)} \log{\left(- \frac{1}{16} + \frac{\sqrt{129}}{16} - \frac{\sqrt{2} i \sqrt{\sqrt{129} + 63}}{16} \right)} \log{\left(- \frac{\sqrt{129}}{16} - \frac{1}{16} - \frac{\sqrt{2} i \sqrt{63 - \sqrt{129}}}{16} \right)} \log{\left(- \frac{\sqrt{129}}{16} - \frac{1}{16} + \frac{\sqrt{2} i \sqrt{63 - \sqrt{129}}}{16} \right)}$$
log(-1/16 - sqrt(129)/16 - i*sqrt(2)*sqrt(63 - sqrt(129))/16)*log(-1/16 - sqrt(129)/16 + i*sqrt(2)*sqrt(63 - sqrt(129))/16)*log(-1/16 + sqrt(129)/16 + sqrt(2)*sqrt(-63 - sqrt(129))/16)*log(-1/16 + sqrt(129)/16 - i*sqrt(2)*sqrt(63 + sqrt(129))/16)
Respuesta numérica
[src]
x1 = -16.3962057587854
x2 = -69.9817144660805
x3 = 99.6642888277683
x4 = -27.5860913914717
x5 = -91.7944294449404
x6 = 82.5480850804397
x7 = 66.6616882162221
x8 = -40.1524620058309
x9 = 98.07761475212
x10 = -11.6996945272541
x11 = 25.9994173158234
x12 = 27.5860913914717
x13 = -60.3785029090425
x14 = -25.9994173158234
x15 = 69.9817144660805
x16 = 33.8692766986513
x17 = -32.282602623003
x18 = -46.4356473130105
x19 = 41.5289469875037
x20 = -41.5289469875037
x21 = 90.4179444632676
x22 = -10.1130204516058
x23 = -2.45335016275338
x24 = -99.6642888277683
x25 = 52.7188326201901
x26 = 11.6996945272541
x27 = -79.2280588305812
x28 = 22.679391065965
x29 = 77.8515738489084
x30 = 47.8121322946833
x31 = 84.134759156088
x32 = -55.6819916775112
x33 = 96.7011297704472
x34 = -13.4330467014642
x35 = -17.9828798344337
x36 = -5.41650922007454
x37 = 74.53154759905
x38 = 1138.12321668661
x39 = -76.2648997732601
x40 = 55.6819916775112
x41 = 61.9651769846908
x42 = -21.3029060842921
x43 = 2.45335016275338
x44 = -33.8692766986513
x45 = -85.5112441377608
x46 = -49.3988063703316
x47 = 8.73653546993297
x48 = 85.5112441377608
x49 = 24.2660651416133
x50 = 54.0953176018629
x51 = -71.5683885417288
x52 = 87.0979182134092
x53 = -98.07761475212
x54 = 40.1524620058309
x55 = 76.2648997732601
x56 = 7.14986139428463
x57 = 68.2483622918704
x58 = -90.4179444632676
x59 = -54.0953176018629
x60 = 32.282602623003
x61 = -24.2660651416133
x62 = 3.8298351444262
x63 = -3.8298351444262
x64 = -84.134759156088
x65 = 30.5492504487929
x66 = -19.7162320086438
x67 = 17.9828798344337
x68 = -47.8121322946833
x69 = 91.7944294449404
x70 = -77.8515738489084
x71 = 71.5683885417288
x72 = -35.2457616803241
x73 = 63.6985291589009
x74 = 38.5657879301826
x75 = -61.9651769846908
x76 = -82.5480850804397
x77 = -63.6985291589009
x78 = -93.3811035205888
x79 = 10.1130204516058
x80 = 19.7162320086438
x81 = 60.3785029090425
x82 = -44.8489732373622
x83 = 16.3962057587854
x84 = -38.5657879301826
x85 = -68.2483622918704
x86 = 46.4356473130105
x87 = -57.4153438517213
x88 = -43.1156210631521
x88 = -43.1156210631521