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La ecuación:
Ecuación x-y=1
Ecuación 7+8x=-2x-5
Ecuación x^2+3x+2=0
Ecuación 3x+3=5x
Expresar {x} en función de y en la ecuación:
18*x-11*y=2
-15*x+17*y=13
4*x-7*y=4
-14*x+10*y=19
Expresiones idénticas
dos *sin(x)-cos(x)^(dos)= dos
2 multiplicar por seno de (x) menos coseno de (x) en el grado (2) es igual a 2
dos multiplicar por seno de (x) menos coseno de (x) en el grado (dos) es igual a dos
2*sin(x)-cos(x)(2)=2
2*sinx-cosx2=2
2sin(x)-cos(x)^(2)=2
2sin(x)-cos(x)(2)=2
2sinx-cosx2=2
2sinx-cosx^2=2
Expresiones semejantes
2*sin(x)+cos(x)^(2)=2
2*sinx-cosx^(2)=2
Expresiones con funciones
Seno sin
sin(x)=-3/4
sin(x)=-1/5
sin2x=0
sin(3*x-pi/6)=1/2
sin(x)^3+cos(x)^3=1
Coseno cos
cos(x)=(-1/2)
cos(x)/5=-1
cos(-x)=-1
cos(2x)=1/2
cos(4*x)=1

2*sin(x)-cos(x)^(2)=2 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

Solución

Ha introducido [src]

              2       
2*sin(x) - cos (x) = 2

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

Simplificar

Solución detallada

Tenemos la ecuación

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

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)

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

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{\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)}

Gráfica

Trazar el gráfico f1(x)

Trazar el gráfico f2(x)

Suma y producto de raíces [src]

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(- 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(\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)

         /    /          ___\\       /    /          ___\\         /    /          ___\\         /    /          ___\\
pi       |    |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|- + ---------||
2        \    \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)} + \frac{\pi}{2} - 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 [src]

     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 [src]

x1 = 70.6858344904728

x2 = 26.7035373336468

x3 = -10.9955745953627

x4 = 14.1371671075501

x5 = -17.2787593377462

x6 = -4.71238930844428

x7 = 76.9690197124121

x8 = 171.216799295141

x9 = -4.71238866775357

x10 = 32.9867231975632

x11 = -36.1283151151309

x12 = 39.2699086832242

x13 = -67.5442421802921

x14 = -67.5442421691475

x15 = -86.393798170863

x16 = -80.1106121881671

x17 = -36.1283154182795

x18 = 89.5353908726546

x19 = 51.8362790471117

x20 = 45.553093716252

x21 = -17.2787598230295

x22 = -61.2610568286838

x23 = 1.57079655968368

x24 = 64.4026490950741

x25 = 26.7035378010916

x26 = -98.9601689022455

x27 = -92.6769829755753

x28 = 1.57079609922276

x29 = -23.5619450830134

x30 = 39.2699078406428

x31 = -73.8274272799426

x32 = -29.8451300959241

x33 = -92.6769836053447

x34 = 76.969020346364

x35 = 20.4203521487293

x36 = 95.818576136834

x37 = 83.2522056392119

x38 = 67690.3261102088

x39 = -48.6946858215598

x40 = 51.8362789015935

x41 = 32.9867225584163

x42 = -10.9955739516139

x43 = 95.818576061021

x44 = -36.1283157628875

x45 = -80.1106125788538

x46 = 70.6858349387112

x47 = -36.1283155336728

x48 = 7.85398196313715

x49 = 7.8539817418448

x50 = -48.6946864571147

x51 = -42.4115010366024

x52 = -54.9778717489189

x53 = -54.9778711011495

x54 = 20.4203520088578

x55 = -61.2610569795535

x56 = 39.2699084858386

x57 = 89.5353903749926

x58 = 89.5353909451703

x59 = 83.252204990125

x60 = 64.4026493079423

x61 = 58.1194644119201

x62 = -61.2610564776906

x63 = -29.8451299456132

x64 = -98.9601682510278

x65 = -42.411500597049

x66 = -86.3937977539453

x67 = -23.5619450102959

x68 = 45.5530932364647

x68 = 45.5530932364647

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¿Cómo usar?

Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

2*sin(x)-cos(x)^(2)=2 la ecuación

Solución numérica:

Solución