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Expresiones semejantes
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Expresiones con funciones
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- cos(3*x)=1
- cos(x)=-3/2
- cos(x)=1/2
log(13)*2*cos(x)^(2)+3*cos(x)-1=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 log(13)*2*cos (x) + 3*cos(x) - 1 = 0
$$\left(2 \log{\left(13 \right)} \cos^{2}{\left(x \right)} + 3 \cos{\left(x \right)}\right) - 1 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(2 \log{\left(13 \right)} \cos^{2}{\left(x \right)} + 3 \cos{\left(x \right)}\right) - 1 = 0$$
cambiamos
$$\log{\left(169 \right)} \cos^{2}{\left(x \right)} + 3 \cos{\left(x \right)} - 1 = 0$$
$$\left(2 \log{\left(13 \right)} \cos^{2}{\left(x \right)} + 3 \cos{\left(x \right)}\right) - 1 = 0$$
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
$$a = 2 \log{\left(13 \right)}$$
$$b = 3$$
$$c = -1$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = \frac{-3 + \sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}}$$
$$w_{2} = \frac{- \sqrt{9 + 8 \log{\left(13 \right)}} - 3}{4 \log{\left(13 \right)}}$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{-3 + \sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{-3 + \sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(\frac{- \sqrt{9 + 8 \log{\left(13 \right)}} - 3}{4 \log{\left(13 \right)}} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(\frac{- \sqrt{9 + 8 \log{\left(13 \right)}} - 3}{4 \log{\left(13 \right)}} \right)}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{-3 + \sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)}$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{-3 + \sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(\frac{- \sqrt{9 + 8 \log{\left(13 \right)}} - 3}{4 \log{\left(13 \right)}} \right)}$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(\frac{- \sqrt{9 + 8 \log{\left(13 \right)}} - 3}{4 \log{\left(13 \right)}} \right)}$$
$$\left(2 \log{\left(13 \right)} \cos^{2}{\left(x \right)} + 3 \cos{\left(x \right)}\right) - 1 = 0$$
cambiamos
$$\log{\left(169 \right)} \cos^{2}{\left(x \right)} + 3 \cos{\left(x \right)} - 1 = 0$$
$$\left(2 \log{\left(13 \right)} \cos^{2}{\left(x \right)} + 3 \cos{\left(x \right)}\right) - 1 = 0$$
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
$$a = 2 \log{\left(13 \right)}$$
$$b = 3$$
$$c = -1$$
, entonces
D = b^2 - 4 * a * c =
(3)^2 - 4 * (2*log(13)) * (-1) = 9 + 8*log(13)
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = \frac{-3 + \sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}}$$
$$w_{2} = \frac{- \sqrt{9 + 8 \log{\left(13 \right)}} - 3}{4 \log{\left(13 \right)}}$$
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:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{-3 + \sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{-3 + \sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(\frac{- \sqrt{9 + 8 \log{\left(13 \right)}} - 3}{4 \log{\left(13 \right)}} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(\frac{- \sqrt{9 + 8 \log{\left(13 \right)}} - 3}{4 \log{\left(13 \right)}} \right)}$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{-3 + \sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)}$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{-3 + \sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(\frac{- \sqrt{9 + 8 \log{\left(13 \right)}} - 3}{4 \log{\left(13 \right)}} \right)}$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(\frac{- \sqrt{9 + 8 \log{\left(13 \right)}} - 3}{4 \log{\left(13 \right)}} \right)}$$
Respuesta rápida
[src]
/ _______________\
| 3 \/ 9 + 8*log(13) |
x1 = - acos|- --------- - -----------------| + 2*pi
\ 4*log(13) 4*log(13) /
$$x_{1} = - \operatorname{acos}{\left(- \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} - \frac{3}{4 \log{\left(13 \right)}} \right)} + 2 \pi$$
/ _______________\
| 3 \/ 9 + 8*log(13) |
x2 = - acos|- --------- + -----------------| + 2*pi
\ 4*log(13) 4*log(13) /
$$x_{2} = - \operatorname{acos}{\left(- \frac{3}{4 \log{\left(13 \right)}} + \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)} + 2 \pi$$
/ _______________\
| 3 \/ 9 + 8*log(13) |
x3 = acos|- --------- - -----------------|
\ 4*log(13) 4*log(13) /
$$x_{3} = \operatorname{acos}{\left(- \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} - \frac{3}{4 \log{\left(13 \right)}} \right)}$$
/ _______________\
| 3 \/ 9 + 8*log(13) |
x4 = acos|- --------- + -----------------|
\ 4*log(13) 4*log(13) /
$$x_{4} = \operatorname{acos}{\left(- \frac{3}{4 \log{\left(13 \right)}} + \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)}$$
x4 = acos(-3/(4*log(13)) + sqrt(9 + 8*log(13))/(4*log(13)))
Suma y producto de raíces
[src]
suma
/ _______________\ / _______________\ / _______________\ / _______________\
| 3 \/ 9 + 8*log(13) | | 3 \/ 9 + 8*log(13) | | 3 \/ 9 + 8*log(13) | | 3 \/ 9 + 8*log(13) |
- acos|- --------- - -----------------| + 2*pi + - acos|- --------- + -----------------| + 2*pi + acos|- --------- - -----------------| + acos|- --------- + -----------------|
\ 4*log(13) 4*log(13) / \ 4*log(13) 4*log(13) / \ 4*log(13) 4*log(13) / \ 4*log(13) 4*log(13) /
$$\operatorname{acos}{\left(- \frac{3}{4 \log{\left(13 \right)}} + \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)} + \left(\operatorname{acos}{\left(- \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} - \frac{3}{4 \log{\left(13 \right)}} \right)} + \left(\left(- \operatorname{acos}{\left(- \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} - \frac{3}{4 \log{\left(13 \right)}} \right)} + 2 \pi\right) + \left(- \operatorname{acos}{\left(- \frac{3}{4 \log{\left(13 \right)}} + \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)} + 2 \pi\right)\right)\right)$$
=
4*pi
$$4 \pi$$
producto
/ / _______________\ \ / / _______________\ \ / _______________\ / _______________\ | | 3 \/ 9 + 8*log(13) | | | | 3 \/ 9 + 8*log(13) | | | 3 \/ 9 + 8*log(13) | | 3 \/ 9 + 8*log(13) | |- acos|- --------- - -----------------| + 2*pi|*|- acos|- --------- + -----------------| + 2*pi|*acos|- --------- - -----------------|*acos|- --------- + -----------------| \ \ 4*log(13) 4*log(13) / / \ \ 4*log(13) 4*log(13) / / \ 4*log(13) 4*log(13) / \ 4*log(13) 4*log(13) /
$$\left(- \operatorname{acos}{\left(- \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} - \frac{3}{4 \log{\left(13 \right)}} \right)} + 2 \pi\right) \left(- \operatorname{acos}{\left(- \frac{3}{4 \log{\left(13 \right)}} + \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)} + 2 \pi\right) \operatorname{acos}{\left(- \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} - \frac{3}{4 \log{\left(13 \right)}} \right)} \operatorname{acos}{\left(- \frac{3}{4 \log{\left(13 \right)}} + \frac{\sqrt{9 + 8 \log{\left(13 \right)}}}{4 \log{\left(13 \right)}} \right)}$$
=
/ / ____________________\ \ / / ____________________\ \ / ____________________\ / ____________________\ | |-3 + \/ 9 + log(815730721) | | | |-3 - \/ 9 + log(815730721) | | |-3 + \/ 9 + log(815730721) | |-3 - \/ 9 + log(815730721) | |- acos|---------------------------| + 2*pi|*|- acos|---------------------------| + 2*pi|*acos|---------------------------|*acos|---------------------------| \ \ 4*log(13) / / \ \ 4*log(13) / / \ 4*log(13) / \ 4*log(13) /
$$\left(- \operatorname{acos}{\left(\frac{-3 + \sqrt{9 + \log{\left(815730721 \right)}}}{4 \log{\left(13 \right)}} \right)} + 2 \pi\right) \left(- \operatorname{acos}{\left(\frac{- \sqrt{9 + \log{\left(815730721 \right)}} - 3}{4 \log{\left(13 \right)}} \right)} + 2 \pi\right) \operatorname{acos}{\left(\frac{-3 + \sqrt{9 + \log{\left(815730721 \right)}}}{4 \log{\left(13 \right)}} \right)} \operatorname{acos}{\left(\frac{- \sqrt{9 + \log{\left(815730721 \right)}} - 3}{4 \log{\left(13 \right)}} \right)}$$
(-acos((-3 + sqrt(9 + log(815730721)))/(4*log(13))) + 2*pi)*(-acos((-3 - sqrt(9 + log(815730721)))/(4*log(13))) + 2*pi)*acos((-3 + sqrt(9 + log(815730721)))/(4*log(13)))*acos((-3 - sqrt(9 + log(815730721)))/(4*log(13)))
Respuesta numérica
[src]
x1 = -33.951575403966
x2 = 30.0845696266773
x3 = -17.5181990123182
x4 = -91.7121307396257
x5 = -54.0130188965482
x6 = -26.4640981379389
x7 = -3.74753643911152
x8 = 128.19935501166
x9 = 84.2170578614027
x10 = -11.2350137051386
x11 = -92.9164226984732
x12 = 61.5004961625753
x13 = -30.0845696266773
x14 = -39.0304687522981
x15 = 99.1996080056528
x16 = 13.8977275235798
x17 = -41.446648282189
x18 = -99.1996080056528
x19 = -60.2962042037278
x20 = 16.3139070534707
x21 = 32.7472834451185
x22 = 40.2347607111456
x23 = 26.4640981379389
x24 = -61.5004961625753
x25 = 27.6683900967864
x26 = 97.9953160468053
x27 = 66.5793895109074
x28 = 22.5970923606503
x29 = -80.350052084114
x30 = -70.4463952881961
x31 = 3.74753643911152
x32 = -55.2173108553957
x33 = 57.8800246738369
x34 = -57.8800246738369
x35 = 11.2350137051386
x36 = 70.4463952881961
x37 = -64.1632099810165
x38 = 46.5179460183252
x39 = 418.437766712964
x40 = -151226.239622066
x41 = 80.350052084114
x42 = -51.5968393666573
x43 = 55.2173108553957
x44 = -32.7472834451185
x45 = -40.2347607111456
x46 = -21.3852047896068
x47 = 64.1632099810165
x48 = 36.3677549338569
x49 = 39.0304687522981
x50 = -97.9953160468053
x51 = -10.0307217462911
x52 = 74.0668667769344
x53 = 76.7295805953756
x54 = -4.95182839795898
x55 = -27.6683900967864
x56 = 2.53564886806806
x57 = -13.8977275235798
x58 = 77.9338725542231
x59 = -74.0668667769344
x60 = -95.5791365169144
x61 = -48.9341255482161
x62 = 10.0307217462911
x63 = -77.9338725542231
x64 = -47.7298335893686
x65 = 91.7121307396257
x66 = -76.7295805953756
x67 = -67.7836814697549
x68 = 60.2962042037278
x69 = -71.6506872470435
x70 = 90.5002431685823
x71 = 23.8013843194977
x72 = -16.3139070534707
x73 = -7.61454221640019
x74 = -20.1809128307594
x75 = 67.7836814697549
x76 = 54.0130188965482
x77 = 17.5181990123182
x78 = 20.1809128307594
x79 = -23.8013843194977
x80 = 71.6506872470435
x81 = -84.2170578614027
x82 = 83.0127659025552
x83 = -85.4289454324461
x84 = -90.5002431685823
x85 = 47.7298335893686
x86 = 33.951575403966
x86 = 33.951575403966