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La ecuación:
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Expresiones con funciones
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sin(2*x-pi/3)=0

2*cos^2(x)+sin(x)+1=0 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

Solución

Ha introducido [src]

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

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

Simplificar

Solución detallada

Tenemos la ecuación

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

cambiamos

\sin{\left(x \right)} + \cos{\left(2 x \right)} + 2 = 0

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

b = 1

c = 3

, entonces

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

(1)^2 - 4 * (-2) * (3) = 25

Como D > 0 la ecuación tiene dos raíces.

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

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

w_{1} = -1

w_{2} = \frac{3}{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

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(\frac{3}{2} \right)}

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

x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(\frac{3}{2} \right)}

Gráfica

Trazar el gráfico f1(x)

Trazar el gráfico f2(x)

Respuesta rápida [src]

     -pi 
x1 = ----
      2

x_{1} = - \frac{\pi}{2}

         /    /        ___\\         /    /        ___\\
         |    |2   I*\/ 5 ||         |    |2   I*\/ 5 ||
x2 = 2*re|atan|- - -------|| + 2*I*im|atan|- - -------||
         \    \3      3   //         \    \3      3   //

x_{2} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{2}{3} - \frac{\sqrt{5} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{2}{3} - \frac{\sqrt{5} i}{3} \right)}\right)}

         /    /        ___\\         /    /        ___\\
         |    |2   I*\/ 5 ||         |    |2   I*\/ 5 ||
x3 = 2*re|atan|- + -------|| + 2*I*im|atan|- + -------||
         \    \3      3   //         \    \3      3   //

x_{3} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}\right)}

x3 = 2*re(atan(2/3 + sqrt(5)*i/3)) + 2*i*im(atan(2/3 + sqrt(5)*i/3))

Suma y producto de raíces [src]

suma

           /    /        ___\\         /    /        ___\\       /    /        ___\\         /    /        ___\\
  pi       |    |2   I*\/ 5 ||         |    |2   I*\/ 5 ||       |    |2   I*\/ 5 ||         |    |2   I*\/ 5 ||
- -- + 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{2}{3} - \frac{\sqrt{5} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{2}{3} - \frac{\sqrt{5} i}{3} \right)}\right)}\right)\right) + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}\right)}\right)

    /    /        ___\\       /    /        ___\\              /    /        ___\\         /    /        ___\\
    |    |2   I*\/ 5 ||       |    |2   I*\/ 5 ||   pi         |    |2   I*\/ 5 ||         |    |2   I*\/ 5 ||
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{2}{3} - \frac{\sqrt{5} i}{3} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{2}{3} - \frac{\sqrt{5} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}\right)}

producto

     /    /    /        ___\\         /    /        ___\\\ /    /    /        ___\\         /    /        ___\\\
-pi  |    |    |2   I*\/ 5 ||         |    |2   I*\/ 5 ||| |    |    |2   I*\/ 5 ||         |    |2   I*\/ 5 |||
----*|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{2}{3} - \frac{\sqrt{5} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{2}{3} - \frac{\sqrt{5} i}{3} \right)}\right)}\right) \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}\right)}\right)

      /    /    /        ___\\     /    /        ___\\\ /    /    /        ___\\     /    /        ___\\\
      |    |    |2   I*\/ 5 ||     |    |2   I*\/ 5 ||| |    |    |2   I*\/ 5 ||     |    |2   I*\/ 5 |||
-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{2}{3} - \frac{\sqrt{5} i}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{2}{3} - \frac{\sqrt{5} i}{3} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}\right)}\right)

-2*pi*(i*im(atan(2/3 - i*sqrt(5)/3)) + re(atan(2/3 - i*sqrt(5)/3)))*(i*im(atan(2/3 + i*sqrt(5)/3)) + re(atan(2/3 + i*sqrt(5)/3)))

Respuesta numérica [src]

x1 = -139.800873249665

x2 = -14.1371668388268

x3 = -89.535390747472

x4 = 36.1283156694962

x5 = -76.9690202883133

x6 = 4.71238876303057

x7 = -70.6858348853437

x8 = 67.5442425428501

x9 = 92.676983075166

x10 = 92.6769826279201

x11 = -95.8185758681129

x12 = -20.420352026649

x13 = 17.2787593859766

x14 = -39.2699083927222

x15 = 48.6946859189952

x16 = 86.3937984757401

x17 = 98.9601687890347

x18 = 48.6946855745421

x19 = 23.5619454986408

x20 = -83.2522052663543

x21 = 29.8451303209123

x22 = 67.5442422847282

x23 = 86.3937983369643

x24 = -83.2522055482568

x25 = 54.9778716517999

x26 = 4.71238853044362

x27 = 42.4115007288249

x28 = 73.8274274802772

x29 = -32.9867231376832

x30 = -76.969019769254

x31 = -58.119464161524

x32 = -20.4203515680363

x33 = -64.402649182671

x34 = 23.5619451291519

x35 = -7.85398149817977

x36 = -83.2522055805072

x37 = -51.8362786896646

x38 = 4087901.62190845

x39 = 61.2610565245164

x40 = 86.3937978948071

x41 = -26.7035372754773

x42 = -14.1371675548369

x43 = -39.2699085196455

x44 = 10.9955745128692

x45 = -102.101761152532

x46 = -32.9867226275421

x47 = 61.2610570268883

x48 = 54.9778711648428

x49 = -70.6858349737007

x50 = -1.57079643014235

x51 = 86.3937978880117

x52 = -45.5530935889045

x53 = 10.9955740133772

x54 = 98.9601683166916

x55 = -58.1194639990021

x56 = 17.278759875892

x57 = -70.6858344272025

x58 = -1.57079608708683

x59 = -26.703537750735

x60 = -64.4026487407202

x61 = 80.1106131421182

x61 = 80.1106131421182

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Expresiones con funciones

2*cos^2(x)+sin(x)+1=0 la ecuación

Solución numérica:

Solución