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- La ecuación:
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Ecuación 0*x=0
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Ecuación z^3-1=0
- Ecuación x^4-50*x^2+49=0
- Ecuación x-y=3
- Expresar {x} en función de y en la ecuación:
- -4*x+18*y=-14
- 18*x-12*y=-20
- -1*x-17*y=-12
- -6*x+1*y=14
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Expresiones idénticas
- dos (tgx-ctgx)= tres ^ uno / dos ((tgx)^ dos +(ctgx)^ dos)- dos * tres ^ uno / dos
- 2(tgx menos ctgx) es igual a 3 en el grado 1 dividir por 2((tgx) al cuadrado más (ctgx) al cuadrado ) menos 2 multiplicar por 3 en el grado 1 dividir por 2
- dos (tgx menos ctgx) es igual a tres en el grado uno dividir por dos ((tgx) en el grado dos más (ctgx) en el grado dos) menos dos multiplicar por tres en el grado uno dividir por dos
- 2(tgx-ctgx)=31/2((tgx)2+(ctgx)2)-2*31/2
- 2tgx-ctgx=31/2tgx2+ctgx2-2*31/2
- 2(tgx-ctgx)=3^1/2((tgx)²+(ctgx)²)-2*3^1/2
- 2(tgx-ctgx)=3 en el grado 1/2((tgx) en el grado 2+(ctgx) en el grado 2)-2*3 en el grado 1/2
- 2(tgx-ctgx)=3^1/2((tgx)^2+(ctgx)^2)-23^1/2
- 2(tgx-ctgx)=31/2((tgx)2+(ctgx)2)-231/2
- 2tgx-ctgx=31/2tgx2+ctgx2-231/2
- 2tgx-ctgx=3^1/2tgx^2+ctgx^2-23^1/2
- 2(tgx-ctgx)=3^1 dividir por 2((tgx)^2+(ctgx)^2)-2*3^1 dividir por 2
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Expresiones semejantes
- 2(tgx-ctgx)=3^1/2((tgx)^2-(ctgx)^2)-2*3^1/2
- 2(tgx+ctgx)=3^1/2((tgx)^2+(ctgx)^2)-2*3^1/2
- 2(tgx-ctgx)=3^1/2((tgx)^2+(ctgx)^2)+2*3^1/2
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Expresiones con funciones
- tgx
- tgx/3=sqrt(3)/3
- tgx=-1
- tgx =2sinx
- tgx=tg10x
- tgx-3^(1/2)*ctgx=3^(1/2)-1
- ctgx
- ctgx=19/15
- ctgx=-1
- ctgx/3-1=0
- ctgx+(sinx)/(1+cosx)=2
- ctgx*sqrt(tgx)=0
- tgx
- tgx/3=sqrt(3)/3
- tgx=-1
- tgx =2sinx
- tgx=tg10x
- tgx-3^(1/2)*ctgx=3^(1/2)-1
- ctgx
- ctgx=19/15
- ctgx=-1
- ctgx/3-1=0
- ctgx+(sinx)/(1+cosx)=2
- ctgx*sqrt(tgx)=0
2(tgx-ctgx)=3^1/2((tgx)^2+(ctgx)^2)-2*3^1/2 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
___ / 2 2 \ ___ 2*(tan(x) - cot(x)) = \/ 3 *\tan (x) + cot (x)/ - 2*\/ 3
$$2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right) = \sqrt{3} \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)}\right) - 2 \sqrt{3}$$
Solución detallada
Tenemos la ecuación
$$2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right) = \sqrt{3} \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)}\right) - 2 \sqrt{3}$$
cambiamos
$$- \sqrt{3} \left(\tan^{2}{\left(x \right)} + \frac{1}{\tan^{2}{\left(x \right)}}\right) + 2 \tan{\left(x \right)} - 2 \cot{\left(x \right)} - 1 + 2 \sqrt{3} = 0$$
$$2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right) - \sqrt{3} \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)}\right) - 1 + 2 \sqrt{3} = 0$$
Sustituimos
$$w = \cot{\left(x \right)}$$
Abramos la expresión en la ecuación
$$- 2 w - \sqrt{3} \left(w^{2} + \tan^{2}{\left(x \right)}\right) + 2 \tan{\left(x \right)} - 1 + 2 \sqrt{3} = 0$$
Obtenemos la ecuación cuadrática
$$- \sqrt{3} w^{2} - 2 w - \sqrt{3} \tan^{2}{\left(x \right)} + 2 \tan{\left(x \right)} - 1 + 2 \sqrt{3} = 0$$
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
$$a = - \sqrt{3}$$
$$b = -2$$
$$c = - \sqrt{3} \tan^{2}{\left(x \right)} + 2 \tan{\left(x \right)} - 1 + 2 \sqrt{3}$$
, entonces
La ecuación tiene dos raíces.
o
$$w_{1} = - \frac{\sqrt{3} \left(\sqrt{4 \sqrt{3} \left(- \sqrt{3} \tan^{2}{\left(x \right)} + 2 \tan{\left(x \right)} - 1 + 2 \sqrt{3}\right) + 4} + 2\right)}{6}$$
$$w_{2} = - \frac{\sqrt{3} \left(2 - \sqrt{4 \sqrt{3} \left(- \sqrt{3} \tan^{2}{\left(x \right)} + 2 \tan{\left(x \right)} - 1 + 2 \sqrt{3}\right) + 4}\right)}{6}$$
hacemos cambio inverso
$$\cot{\left(x \right)} = w$$
sustituimos w:
$$2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right) = \sqrt{3} \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)}\right) - 2 \sqrt{3}$$
cambiamos
$$- \sqrt{3} \left(\tan^{2}{\left(x \right)} + \frac{1}{\tan^{2}{\left(x \right)}}\right) + 2 \tan{\left(x \right)} - 2 \cot{\left(x \right)} - 1 + 2 \sqrt{3} = 0$$
$$2 \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right) - \sqrt{3} \left(\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)}\right) - 1 + 2 \sqrt{3} = 0$$
Sustituimos
$$w = \cot{\left(x \right)}$$
Abramos la expresión en la ecuación
$$- 2 w - \sqrt{3} \left(w^{2} + \tan^{2}{\left(x \right)}\right) + 2 \tan{\left(x \right)} - 1 + 2 \sqrt{3} = 0$$
Obtenemos la ecuación cuadrática
$$- \sqrt{3} w^{2} - 2 w - \sqrt{3} \tan^{2}{\left(x \right)} + 2 \tan{\left(x \right)} - 1 + 2 \sqrt{3} = 0$$
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
$$a = - \sqrt{3}$$
$$b = -2$$
$$c = - \sqrt{3} \tan^{2}{\left(x \right)} + 2 \tan{\left(x \right)} - 1 + 2 \sqrt{3}$$
, entonces
D = b^2 - 4 * a * c =
(-2)^2 - 4 * (-sqrt(3)) * (-1 + 2*sqrt(3) + 2*tan(x) - sqrt(3)*tan(x)^2) = 4 + 4*sqrt(3)*(-1 + 2*sqrt(3) + 2*tan(x) - sqrt(3)*tan(x)^2)
La ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = - \frac{\sqrt{3} \left(\sqrt{4 \sqrt{3} \left(- \sqrt{3} \tan^{2}{\left(x \right)} + 2 \tan{\left(x \right)} - 1 + 2 \sqrt{3}\right) + 4} + 2\right)}{6}$$
$$w_{2} = - \frac{\sqrt{3} \left(2 - \sqrt{4 \sqrt{3} \left(- \sqrt{3} \tan^{2}{\left(x \right)} + 2 \tan{\left(x \right)} - 1 + 2 \sqrt{3}\right) + 4}\right)}{6}$$
hacemos cambio inverso
$$\cot{\left(x \right)} = w$$
sustituimos w:
Suma y producto de raíces
[src]
suma
pi pi pi pi - -- - -- + -- + -- 4 6 4 3
$$\left(\left(- \frac{\pi}{4} - \frac{\pi}{6}\right) + \frac{\pi}{4}\right) + \frac{\pi}{3}$$
=
pi -- 6
$$\frac{\pi}{6}$$
producto
-pi -pi pi pi ----*----*--*-- 4 6 4 3
$$\frac{\pi}{3} \frac{\pi}{4} \cdot - \frac{\pi}{4} \left(- \frac{\pi}{6}\right)$$
=
4 pi --- 288
$$\frac{\pi^{4}}{288}$$
pi^4/288
Respuesta rápida
[src]
-pi
x1 = ----
4
$$x_{1} = - \frac{\pi}{4}$$
-pi
x2 = ----
6
$$x_{2} = - \frac{\pi}{6}$$
pi
x3 = --
4
$$x_{3} = \frac{\pi}{4}$$
pi
x4 = --
3
$$x_{4} = \frac{\pi}{3}$$
x4 = pi/3
Respuesta numérica
[src]
x1 = 55.7632696012188
x2 = -33.7721210260903
x3 = -43.1968989868597
x4 = -47.9092879672443
x5 = 46.3384916404494
x6 = 16.4933614313464
x7 = 90.3207887907066
x8 = 70.162235930172
x9 = 60.4756585816035
x10 = -99.7455667514759
x11 = -8.37758040957278
x12 = -96.342174710087
x13 = -85.6083998103219
x14 = -55.7632696012188
x15 = 32.2013246992954
x16 = 18.0641577581413
x17 = 35.6047167406843
x18 = -46.0766922526503
x19 = 63.8790506229925
x20 = -91.8915851175014
x21 = 41.8879020478639
x22 = -82.2050077689329
x23 = 48.1710873550435
x24 = -13.3517687777566
x25 = 13.6135681655558
x26 = -3.92699081698724
x27 = -71.4712328691678
x28 = -97.9129710368819
x29 = 56.025068989018
x30 = -75.9218224617533
x31 = 40.0553063332699
x32 = -25.9181393921158
x33 = -31.9395253114962
x34 = 5.75958653158129
x35 = 2.35619449019234
x36 = 85.870199198121
x37 = 96.6039740978861
x38 = -11.7809724509617
x39 = -16.2315620435473
x40 = 4.18879020478639
x41 = 82.4668071567321
x42 = -74.3510261349584
x43 = 54.1924732744239
x44 = -17.8023583703422
x45 = -49.4800842940392
x46 = 34.0339204138894
x47 = -2.0943951023932
x48 = 12.0427718387609
x49 = 49.7418836818384
x50 = 84.037603483527
x51 = 88.7499924639117
x52 = -77.7544181763474
x53 = 24.3473430653209
x54 = -87.1791961371168
x55 = 27.7507351067098
x56 = -24.0855436775217
x57 = 22.776546738526
x58 = 26.1799387799149
x59 = 44.7676953136546
x60 = -9.94837673636768
x61 = 19.8967534727354
x62 = 78.0162175641465
x63 = -57.3340659280137
x64 = -90.0589894029074
x65 = 76.1836218495525
x66 = 8.63937979737193
x67 = -69.9004365423729
x68 = 68.329640215578
x69 = -63.6172512351933
x70 = 98.174770424681
x71 = -19.6349540849362
x72 = -52.3598775598299
x73 = -93.4623814442964
x74 = -41.6261026600648
x75 = -27.4889357189107
x76 = -60.2138591938044
x77 = -61.7846555205993
x78 = 30.6305283725005
x79 = 93.7241808320955
x80 = 10.2101761241668
x81 = -68.0678408277789
x82 = 100.007366139275
x83 = 74.6128255227576
x84 = 81.1578102177363
x85 = -30.3687289847013
x86 = -53.9306738866248
x87 = -65.1880475619882
x88 = -79.3252145031423
x89 = -38.2227106186758
x90 = 52.621676947629
x91 = -83.7758040957278
x92 = 71.733032256967
x93 = 38.484510006475
x94 = 92.1533845053006
x95 = -5.49778714378214
x96 = -35.3429173528852
x97 = 62.3082542961976
x98 = -39.7935069454707
x99 = -21.2057504117311
x100 = 66.7588438887831
x100 = 66.7588438887831