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La ecuación:
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Expresar {x} en función de y en la ecuación:
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Expresiones semejantes
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Expresiones con funciones
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cos(3*x-pi/4)=-1

-3sinx-1+cos^2(x)=0 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

Solución

Ha introducido [src]

                   2       
-3*sin(x) - 1 + cos (x) = 0

\left(- 3 \sin{\left(x \right)} - 1\right) + \cos^{2}{\left(x \right)} = 0

Simplificar

Solución detallada

Tenemos la ecuación

\left(- 3 \sin{\left(x \right)} - 1\right) + \cos^{2}{\left(x \right)} = 0

cambiamos

- \left(\sin{\left(x \right)} + 3\right) \sin{\left(x \right)} = 0

- \sin^{2}{\left(x \right)} - 3 \sin{\left(x \right)} = 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 = -3

c = 0

, entonces

D = b^2 - 4 * a * c =

(-3)^2 - 4 * (-1) * (0) = 9

Como D > 0 la ecuación tiene dos raíces.

w1 = (-b + sqrt(D)) / (2*a)

w2 = (-b - sqrt(D)) / (2*a)

w_{1} = -3

w_{2} = 0

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

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(-3 \right)}

x_{1} = 2 \pi n - \operatorname{asin}{\left(3 \right)}

x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}

x_{2} = 2 \pi n + \operatorname{asin}{\left(0 \right)}

x_{2} = 2 \pi n

x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi

x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(-3 \right)}

x_{3} = 2 \pi n + \pi + \operatorname{asin}{\left(3 \right)}

x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi

x_{4} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi

x_{4} = 2 \pi n + \pi

Gráfica

Trazar el gráfico f1(x)

Trazar el gráfico f2(x)

Respuesta rápida [src]

x1 = 0

x_{1} = 0

           /    /          ___\\         /    /          ___\\
           |    |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))

Suma y producto de raíces [src]

suma

      /    /          ___\\         /    /          ___\\         /    /          ___\\         /    /          ___\\
      |    |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|- + ---------||
      \    \3       3    //         \    \3       3    //         \    \3       3    //         \    \3       3    //

\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*re|atan|- - ---------|| - 2*re|atan|- + ---------|| - 2*I*im|atan|- - ---------|| - 2*I*im|atan|- + ---------||
      \    \3       3    //       \    \3       3    //         \    \3       3    //         \    \3       3    //

- 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

  /      /    /          ___\\         /    /          ___\\\ /      /    /          ___\\         /    /          ___\\\
  |      |    |1   2*I*\/ 2 ||         |    |1   2*I*\/ 2 ||| |      |    |1   2*I*\/ 2 ||         |    |1   2*I*\/ 2 |||
0*|- 2*re|atan|- - ---------|| - 2*I*im|atan|- - ---------|||*|- 2*re|atan|- + ---------|| - 2*I*im|atan|- + ---------|||
  \      \    \3       3    //         \    \3       3    /// \      \    \3       3    //         \    \3       3    ///

0 \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)

0

Respuesta numérica [src]

x1 = 40.8407044966673

x2 = 0.0

x3 = -18.8495559215388

x4 = -56.5486677646163

x5 = 97.3893722612836

x6 = 34.5575191894877

x7 = 53.4070751110265

x8 = 47.1238898038469

x9 = -97.3893722612836

x10 = 62.8318530717959

x11 = 87.9645943005142

x12 = 43.9822971502571

x13 = 37.6991118430775

x14 = -21.9911485751286

x15 = 3.14159265358979

x16 = 65.9734457253857

x17 = 69.1150383789755

x18 = -50.2654824574367

x19 = -94.2477796076938

x20 = -75.398223686155

x21 = -53.4070751110265

x22 = 12.5663706143592

x23 = -9.42477796076938

x24 = -34.5575191894877

x25 = 21.9911485751286

x26 = -47.1238898038469

x27 = -43.9822971502571

x28 = 28.2743338823081

x29 = -31.4159265358979

x30 = -3.14159265358979

x31 = -6.28318530717959

x32 = 216.769893097696

x33 = -25.1327412287183

x34 = 31.4159265358979

x35 = -62.8318530717959

x36 = -65.9734457253857

x37 = 72.2566310325652

x38 = -59.6902604182061

x39 = -12.5663706143592

x40 = 94.2477796076938

x41 = 81.6814089933346

x42 = -91.106186954104

x43 = -100.530964914873

x44 = 59.6902604182061

x45 = -40.8407044966673

x46 = 91.106186954104

x47 = 78.5398163397448

x48 = -106.814150222053

x49 = 56.5486677646163

x50 = 84.8230016469244

x51 = 100.530964914873

x52 = -69.1150383789755

x53 = 9.42477796076938

x54 = -84.8230016469244

x55 = -78.5398163397448

x56 = -87.9645943005142

x57 = -81.6814089933346

x58 = 15.707963267949

x59 = -28.2743338823081

x60 = -15.707963267949

x61 = -37.6991118430775

x62 = 18.8495559215388

x63 = 25.1327412287183

x64 = 50.2654824574367

x65 = -72.2566310325652

x66 = 75.398223686155

x67 = 6.28318530717959

x67 = 6.28318530717959

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Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

-3sinx-1+cos^2(x)=0 la ecuación

Solución numérica:

Solución