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Expresiones semejantes
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Expresiones con funciones
- Seno sin
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- Coseno cos
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- cos(x)/3=0
- cos(2*x)+cos(x)=0
- cos(8*pi*x/6)=sqrt(3)/2
2*sin(x)-cos(x)^(2)=2 la ecuación
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Solución
Ha introducido
[src]
2 2*sin(x) - cos (x) = 2
$$2 \sin{\left(x \right)} - \cos^{2}{\left(x \right)} = 2$$
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
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(-3 \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(3 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{\pi}{2}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(-3 \right)}$$
$$x_{4} = 2 \pi n + \pi + \operatorname{asin}{\left(3 \right)}$$
$$2 \sin{\left(x \right)} - \cos^{2}{\left(x \right)} = 2$$
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)
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(-3 \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(3 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{\pi}{2}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(-3 \right)}$$
$$x_{4} = 2 \pi n + \pi + \operatorname{asin}{\left(3 \right)}$$
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 //
$$\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{3} + \frac{2 \sqrt{2} i}{3} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{3} + \frac{2 \sqrt{2} i}{3} \right)}\right)}\right) + \left(\frac{\pi}{2} + \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{3} - \frac{2 \sqrt{2} i}{3} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{3} - \frac{2 \sqrt{2} i}{3} \right)}\right)}\right)\right)$$
=
/ / ___\\ / / ___\\ / / ___\\ / / ___\\ 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
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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