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Expresiones semejantes
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(3cos^2x+7sinx-5)=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 3*cos (x) + 7*sin(x) - 5 = 0
$$\left(7 \sin{\left(x \right)} + 3 \cos^{2}{\left(x \right)}\right) - 5 = 0$$
Solución detallada
Tenemos la ecuación
$$\left(7 \sin{\left(x \right)} + 3 \cos^{2}{\left(x \right)}\right) - 5 = 0$$
cambiamos
$$- 3 \sin^{2}{\left(x \right)} + 7 \sin{\left(x \right)} - 2 = 0$$
$$- 3 \sin^{2}{\left(x \right)} + 7 \sin{\left(x \right)} - 2 = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
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
$$a = -3$$
$$b = 7$$
$$c = -2$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = \frac{1}{3}$$
$$w_{2} = 2$$
hacemos cambio inverso
$$\sin{\left(x \right)} = w$$
Tenemos la ecuación
$$\sin{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{3} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{3} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{3} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{3} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$\left(7 \sin{\left(x \right)} + 3 \cos^{2}{\left(x \right)}\right) - 5 = 0$$
cambiamos
$$- 3 \sin^{2}{\left(x \right)} + 7 \sin{\left(x \right)} - 2 = 0$$
$$- 3 \sin^{2}{\left(x \right)} + 7 \sin{\left(x \right)} - 2 = 0$$
Sustituimos
$$w = \sin{\left(x \right)}$$
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
$$a = -3$$
$$b = 7$$
$$c = -2$$
, entonces
D = b^2 - 4 * a * c =
(7)^2 - 4 * (-3) * (-2) = 25
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = \frac{1}{3}$$
$$w_{2} = 2$$
hacemos cambio inverso
$$\sin{\left(x \right)} = w$$
Tenemos la ecuación
$$\sin{\left(x \right)} = w$$
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
O
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, donde n es cualquier número entero
sustituimos w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{3} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{3} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{3} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{3} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
Respuesta rápida
[src]
/ / ___\\ / / ___\\
| |1 I*\/ 3 || | |1 I*\/ 3 ||
x1 = 2*re|atan|- - -------|| + 2*I*im|atan|- - -------||
\ \2 2 // \ \2 2 //
$$x_{1} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}$$
/ / ___\\ / / ___\\
| |1 I*\/ 3 || | |1 I*\/ 3 ||
x2 = 2*re|atan|- + -------|| + 2*I*im|atan|- + -------||
\ \2 2 // \ \2 2 //
$$x_{2} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}$$
/ ___\ x3 = 2*atan\3 - 2*\/ 2 /
$$x_{3} = 2 \operatorname{atan}{\left(3 - 2 \sqrt{2} \right)}$$
/ ___\ x4 = 2*atan\3 + 2*\/ 2 /
$$x_{4} = 2 \operatorname{atan}{\left(2 \sqrt{2} + 3 \right)}$$
x4 = 2*atan(2*sqrt(2) + 3)
Suma y producto de raíces
[src]
suma
/ / ___\\ / / ___\\ / / ___\\ / / ___\\
| |1 I*\/ 3 || | |1 I*\/ 3 || | |1 I*\/ 3 || | |1 I*\/ 3 || / ___\ / ___\
2*re|atan|- - -------|| + 2*I*im|atan|- - -------|| + 2*re|atan|- + -------|| + 2*I*im|atan|- + -------|| + 2*atan\3 - 2*\/ 2 / + 2*atan\3 + 2*\/ 2 /
\ \2 2 // \ \2 2 // \ \2 2 // \ \2 2 //
$$2 \operatorname{atan}{\left(2 \sqrt{2} + 3 \right)} + \left(2 \operatorname{atan}{\left(3 - 2 \sqrt{2} \right)} + \left(\left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right) + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right)\right)\right)$$
=
/ / ___\\ / / ___\\ / / ___\\ / / ___\\
/ ___\ / ___\ | |1 I*\/ 3 || | |1 I*\/ 3 || | |1 I*\/ 3 || | |1 I*\/ 3 ||
2*atan\3 - 2*\/ 2 / + 2*atan\3 + 2*\/ 2 / + 2*re|atan|- + -------|| + 2*re|atan|- - -------|| + 2*I*im|atan|- + -------|| + 2*I*im|atan|- - -------||
\ \2 2 // \ \2 2 // \ \2 2 // \ \2 2 //
$$2 \operatorname{atan}{\left(3 - 2 \sqrt{2} \right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + 2 \operatorname{atan}{\left(2 \sqrt{2} + 3 \right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}$$
producto
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ | | |1 I*\/ 3 || | |1 I*\/ 3 ||| | | |1 I*\/ 3 || | |1 I*\/ 3 ||| / ___\ / ___\ |2*re|atan|- - -------|| + 2*I*im|atan|- - -------|||*|2*re|atan|- + -------|| + 2*I*im|atan|- + -------|||*2*atan\3 - 2*\/ 2 /*2*atan\3 + 2*\/ 2 / \ \ \2 2 // \ \2 2 /// \ \ \2 2 // \ \2 2 ///
$$\left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right) \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right) 2 \operatorname{atan}{\left(3 - 2 \sqrt{2} \right)} 2 \operatorname{atan}{\left(2 \sqrt{2} + 3 \right)}$$
=
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ | | |1 I*\/ 3 || | |1 I*\/ 3 ||| | | |1 I*\/ 3 || | |1 I*\/ 3 ||| / ___\ / ___\ 16*|I*im|atan|- + -------|| + re|atan|- + -------|||*|I*im|atan|- - -------|| + re|atan|- - -------|||*atan\3 - 2*\/ 2 /*atan\3 + 2*\/ 2 / \ \ \2 2 // \ \2 2 /// \ \ \2 2 // \ \2 2 ///
$$16 \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right) \operatorname{atan}{\left(3 - 2 \sqrt{2} \right)} \operatorname{atan}{\left(2 \sqrt{2} + 3 \right)}$$
16*(i*im(atan(1/2 + i*sqrt(3)/2)) + re(atan(1/2 + i*sqrt(3)/2)))*(i*im(atan(1/2 - i*sqrt(3)/2)) + re(atan(1/2 - i*sqrt(3)/2)))*atan(3 - 2*sqrt(2))*atan(3 + 2*sqrt(2))
Respuesta numérica
[src]
x1 = -5.94334839772546
x2 = 2.80175574413567
x3 = 21.6513116656744
x4 = 38.0389487525316
x5 = 6.62302221663371
x6 = -91.4460238635581
x7 = -41.1805414061214
x8 = 53.0672382015724
x9 = -37.3592749336234
x10 = 203.863685573882
x11 = 84.4831647374703
x12 = 12.9062075238133
x13 = 69.4548752884296
x14 = 44.3221340597112
x15 = 50.6053193668908
x16 = 46.7840528943928
x17 = -16.0478001774031
x18 = -18.5097190120846
x19 = 27.934496972854
x20 = -24.7929043192642
x21 = 56.8885046740704
x22 = -62.4920161623417
x23 = -9.7646148702235
x24 = -232.138019456191
x25 = 71.9167941231111
x26 = -34.8973560989418
x27 = -12.2265337049051
x28 = -97.7292091707377
x29 = -78.879653249199
x30 = -22.3309854845827
x31 = -100.191128005419
x32 = -68.7752014695213
x33 = -66.3132826348398
x34 = 94.5876165171479
x35 = 0.339836909454122
x36 = 31.7557634453521
x37 = 90.7663500446499
x38 = 25.4725781381725
x39 = 19.1893928309929
x40 = -72.5964679420194
x41 = 100.870801824328
x42 = 97.0495353518295
x43 = -56.2088308551622
x44 = 34.2176822800336
x45 = -47.463726713301
x46 = 82.0212459027887
x47 = -53.7469120204806
x48 = -93.9079426982397
x49 = 75.7380605956092
x50 = 88.3044312099683
x51 = -43.642460240803
x52 = 15.3681263584948
x53 = -81.3415720838805
x54 = -60.0300973276602
x55 = 65.6336088159315
x56 = 78.1999794302907
x57 = 9.08494105131526
x58 = -49.9256455479826
x59 = -87.6247573910601
x60 = 63.17168998125
x61 = -75.0583867767009
x62 = -85.1628385563785
x63 = 59.350423508752
x64 = -28.6141707917623
x65 = -31.0760896264438
x66 = 40.5008675872132
x67 = -3.48142956304392
x68 = -1061.5184800039
x69 = -110.295579785097
x69 = -110.295579785097