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- La ecuación:
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Ecuación x^3+3x^2-x-3=0
-
Ecuación x^3-6x^2+11x-6=0
-
Ecuación a+120=2000/5
-
Ecuación x/4+x=12
- Expresar {x} en función de y en la ecuación:
- 9*x-1*y=17
- 10*x-2*y=11
- -18*x-13*y=2
- -12*x-8*y=13
- Integral de d{x}:
- -2sinx
- Derivada de:
-
-2sinx
- Gráfico de la función y =:
- -2sinx
-
Expresiones idénticas
- dos +сos^2x=-2sinx
- 2 más сos al cuadrado x es igual a menos 2 seno de x
- dos más сos al cuadrado x es igual a menos 2 seno de x
- 2+сos2x=-2sinx
- 2+сos²x=-2sinx
- 2+сos en el grado 2x=-2sinx
-
Expresiones semejantes
- 1-2*sin(x)=0
- 2+сos^2x=+2sinx
- sin(x)^(2)-2*sin(x)-3=0
- 2-сos^2x=-2sinx
- 3*x-2*sin(x)=0.3
2+сos^2x=-2sinx la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 2 + cos (x) = -2*sin(x)
$$\cos^{2}{\left(x \right)} + 2 = - 2 \sin{\left(x \right)}$$
Solución detallada
Tenemos la ecuación
$$\cos^{2}{\left(x \right)} + 2 = - 2 \sin{\left(x \right)}$$
cambiamos
$$- \sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} + 3 = 0$$
$$- \sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} + 3 = 0$$
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
$$a = -1$$
$$b = 2$$
$$c = 3$$
, entonces
Como D > 0 la ecuación tiene dos raíces.
o
$$w_{1} = -1$$
$$w_{2} = 3$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{3 \pi}{2}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}$$
$$\cos^{2}{\left(x \right)} + 2 = - 2 \sin{\left(x \right)}$$
cambiamos
$$- \sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} + 3 = 0$$
$$- \sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} + 3 = 0$$
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
$$a = -1$$
$$b = 2$$
$$c = 3$$
, entonces
D = b^2 - 4 * a * c =
(2)^2 - 4 * (-1) * (3) = 16
Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
o
$$w_{1} = -1$$
$$w_{2} = 3$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{1} = 2 \pi n - \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(3 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{3 \pi}{2}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(3 \right)}$$
Suma y producto de raíces
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suma
/ / ___\\ / / ___\\ / / ___\\ / / ___\\ pi | |1 2*I*\/ 2 || | |1 2*I*\/ 2 || | |1 2*I*\/ 2 || | |1 2*I*\/ 2 || - -- + 2*re|atan|- - ---------|| + 2*I*im|atan|- - ---------|| + 2*re|atan|- + ---------|| + 2*I*im|atan|- + ---------|| 2 \ \3 3 // \ \3 3 // \ \3 3 // \ \3 3 //
$$\left(- \frac{\pi}{2} + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{3} - \frac{2 \sqrt{2} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{3} - \frac{2 \sqrt{2} i}{3} \right)}\right)}\right)\right) + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{3} + \frac{2 \sqrt{2} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{3} + \frac{2 \sqrt{2} i}{3} \right)}\right)}\right)$$
=
/ / ___\\ / / ___\\ / / ___\\ / / ___\\
| |1 2*I*\/ 2 || | |1 2*I*\/ 2 || pi | |1 2*I*\/ 2 || | |1 2*I*\/ 2 ||
2*re|atan|- - ---------|| + 2*re|atan|- + ---------|| - -- + 2*I*im|atan|- - ---------|| + 2*I*im|atan|- + ---------||
\ \3 3 // \ \3 3 // 2 \ \3 3 // \ \3 3 //
$$- \frac{\pi}{2} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{3} - \frac{2 \sqrt{2} i}{3} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{3} + \frac{2 \sqrt{2} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{3} - \frac{2 \sqrt{2} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{3} + \frac{2 \sqrt{2} i}{3} \right)}\right)}$$
producto
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ -pi | | |1 2*I*\/ 2 || | |1 2*I*\/ 2 ||| | | |1 2*I*\/ 2 || | |1 2*I*\/ 2 ||| ----*|2*re|atan|- - ---------|| + 2*I*im|atan|- - ---------|||*|2*re|atan|- + ---------|| + 2*I*im|atan|- + ---------||| 2 \ \ \3 3 // \ \3 3 /// \ \ \3 3 // \ \3 3 ///
$$- \frac{\pi}{2} \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{3} - \frac{2 \sqrt{2} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{3} - \frac{2 \sqrt{2} i}{3} \right)}\right)}\right) \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{3} + \frac{2 \sqrt{2} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{3} + \frac{2 \sqrt{2} i}{3} \right)}\right)}\right)$$
=
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\
| | |1 2*I*\/ 2 || | |1 2*I*\/ 2 ||| | | |1 2*I*\/ 2 || | |1 2*I*\/ 2 |||
-2*pi*|I*im|atan|- - ---------|| + re|atan|- - ---------|||*|I*im|atan|- + ---------|| + re|atan|- + ---------|||
\ \ \3 3 // \ \3 3 /// \ \ \3 3 // \ \3 3 ///
$$- 2 \pi \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{3} - \frac{2 \sqrt{2} i}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{3} - \frac{2 \sqrt{2} i}{3} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{3} + \frac{2 \sqrt{2} i}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{3} + \frac{2 \sqrt{2} i}{3} \right)}\right)}\right)$$
-2*pi*(i*im(atan(1/3 - 2*i*sqrt(2)/3)) + re(atan(1/3 - 2*i*sqrt(2)/3)))*(i*im(atan(1/3 + 2*i*sqrt(2)/3)) + re(atan(1/3 + 2*i*sqrt(2)/3)))
Respuesta rápida
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-pi
x1 = ----
2
$$x_{1} = - \frac{\pi}{2}$$
/ / ___\\ / / ___\\
| |1 2*I*\/ 2 || | |1 2*I*\/ 2 ||
x2 = 2*re|atan|- - ---------|| + 2*I*im|atan|- - ---------||
\ \3 3 // \ \3 3 //
$$x_{2} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{3} - \frac{2 \sqrt{2} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{3} - \frac{2 \sqrt{2} i}{3} \right)}\right)}$$
/ / ___\\ / / ___\\
| |1 2*I*\/ 2 || | |1 2*I*\/ 2 ||
x3 = 2*re|atan|- + ---------|| + 2*I*im|atan|- + ---------||
\ \3 3 // \ \3 3 //
$$x_{3} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{3} + \frac{2 \sqrt{2} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{3} + \frac{2 \sqrt{2} i}{3} \right)}\right)}$$
x3 = 2*re(atan(1/3 + 2*sqrt(2)*i/3)) + 2*i*im(atan(1/3 + 2*sqrt(2)*i/3))
Respuesta numérica
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x1 = 61.2610564153358
x2 = 73.827427591263
x3 = 4.71238875528975
x4 = 23.5619451379884
x5 = 86.3937976360352
x6 = 42.4115005526819
x7 = 48.6946859120413
x8 = 80.1106131400977
x9 = 117.809725233756
x10 = 36.1283157033748
x11 = -32.9867231721691
x12 = -83.2522055577573
x13 = -45.5530936309972
x14 = -76.9690203256113
x15 = -51.8362786895378
x16 = -58.1194639985047
x17 = 48.6946863700612
x18 = -139.800873467486
x19 = 54.9778711353875
x20 = -32.9867225263379
x21 = -164.933614398177
x22 = -26.7035372446302
x23 = -7.85398149759801
x24 = 882955.610865625
x25 = 29.845130504401
x26 = 42.4115009058131
x27 = 67.5442418055755
x28 = 29.8451303217623
x29 = 17.2787592660486
x30 = -14.1371665766936
x31 = 17.2787599090696
x32 = -95.8185758680893
x33 = -14.1371668194089
x34 = -64.4026496039382
x35 = 86.3937978876249
x36 = -20.4203520186622
x37 = -89.5353907308516
x38 = -39.2699076412407
x39 = -14.1371668381663
x40 = 67.5442422944741
x41 = -20.4203524688757
x42 = -26.703537882833
x43 = -64.4026491754768
x44 = 10.9955739814993
x45 = 23.5619446676752
x46 = 73.8274274813446
x47 = -70.6858343985417
x48 = -89.5353907485116
x49 = -108.38494727522
x50 = 4.71238886235498
x51 = -45.5530935897427
x52 = 98.9601682894894
x53 = 10.9955746230205
x54 = -1.57079653638307
x55 = 61.261057062553
x56 = 92.6769830689412
x57 = -39.2699079075865
x58 = 54.9778717720118
x59 = -1.57079643080582
x60 = 42.4115007283113
x61 = 92.6769835070253
x62 = -58.119463652249
x63 = -39.2699084013107
x64 = -83.2522050480464
x65 = 98.9601689206173
x66 = -76.9690196760468
x67 = -70.6858350312864
x68 = -133.517687838813
x69 = 4.71238923181769
x70 = -83.2522051037983
x71 = 212.057503588662
x71 = 212.057503588662