Sr Examen

Otras calculadoras

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

cos(2*x)+3*sin(x)+1=0 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

v

Solución numérica:

Buscar la solución numérica en el intervalo [, ]

Solución

Ha introducido [src] 
cos(2*x) + 3*sin(x) + 1 = 0
(3sin⁡(x)+cos⁡(2x))+1=0\left(3 \sin{\left(x \right)} + \cos{\left(2 x \right)}\right) + 1 = 0
Simplificar
Solución detallada
Tenemos la ecuación
(3sin⁡(x)+cos⁡(2x))+1=0\left(3 \sin{\left(x \right)} + \cos{\left(2 x \right)}\right) + 1 = 0
cambiamos
3sin⁡(x)+cos⁡(2x)+1=03 \sin{\left(x \right)} + \cos{\left(2 x \right)} + 1 = 0
−2sin⁡2(x)+3sin⁡(x)+2=0- 2 \sin^{2}{\left(x \right)} + 3 \sin{\left(x \right)} + 2 = 0
Sustituimos
w=sin⁡(x)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:
w1=D−b2aw_{1} = \frac{\sqrt{D} - b}{2 a}
w2=−D−b2aw_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=−2a = -2
b=3b = 3
c=2c = 2
, entonces
D = b^2 - 4 * a * c = 

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

Como D > 0 la ecuación tiene dos raíces.
w1 = (-b + sqrt(D)) / (2*a)

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

o
w1=−12w_{1} = - \frac{1}{2}
w2=2w_{2} = 2
hacemos cambio inverso
sin⁡(x)=w\sin{\left(x \right)} = w
Tenemos la ecuación
sin⁡(x)=w\sin{\left(x \right)} = w
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
x=2πn+asin⁡(w)x = 2 \pi n + \operatorname{asin}{\left(w \right)}
x=2πn−asin⁡(w)+πx = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
O
x=2πn+asin⁡(w)x = 2 \pi n + \operatorname{asin}{\left(w \right)}
x=2πn−asin⁡(w)+πx = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
, donde n es cualquier número entero
sustituimos w:
x1=2πn+asin⁡(w1)x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}
x1=2πn+asin⁡(−12)x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}
x1=2πn−π6x_{1} = 2 \pi n - \frac{\pi}{6}
x2=2πn+asin⁡(w2)x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}
x2=2πn+asin⁡(2)x_{2} = 2 \pi n + \operatorname{asin}{\left(2 \right)}
x2=2πn+asin⁡(2)x_{2} = 2 \pi n + \operatorname{asin}{\left(2 \right)}
x3=2πn−asin⁡(w1)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
x3=2πn−asin⁡(−12)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi
x3=2πn+7π6x_{3} = 2 \pi n + \frac{7 \pi}{6}
x4=2πn−asin⁡(w2)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi
x4=2πn+π−asin⁡(2)x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}
x4=2πn+π−asin⁡(2)x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}
Gráfica
0-80-60-40-2020406080-1001005-5
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
  5*pi   pi   pi        /      ___\   pi        /      ___\
- ---- - -- + -- - I*log\2 - \/ 3 / + -- - I*log\2 + \/ 3 /
   6     6    2                       2
(π2−ilog⁡(3+2))+((−5π6−π6)+(π2−ilog⁡(2−3)))\left(\frac{\pi}{2} - i \log{\left(\sqrt{3} + 2 \right)}\right) + \left(\left(- \frac{5 \pi}{6} - \frac{\pi}{6}\right) + \left(\frac{\pi}{2} - i \log{\left(2 - \sqrt{3} \right)}\right)\right) 
=
       /      ___\        /      ___\
- I*log\2 + \/ 3 / - I*log\2 - \/ 3 /
−ilog⁡(3+2)−ilog⁡(2−3)- i \log{\left(\sqrt{3} + 2 \right)} - i \log{\left(2 - \sqrt{3} \right)} 
producto
-5*pi -pi  /pi        /      ___\\ /pi        /      ___\\
-----*----*|-- - I*log\2 - \/ 3 /|*|-- - I*log\2 + \/ 3 /|
  6    6   \2                    / \2                    /
−5π6(−π6)(π2−ilog⁡(2−3))(π2−ilog⁡(3+2))- \frac{5 \pi}{6} \left(- \frac{\pi}{6}\right) \left(\frac{\pi}{2} - i \log{\left(2 - \sqrt{3} \right)}\right) \left(\frac{\pi}{2} - i \log{\left(\sqrt{3} + 2 \right)}\right) 
=
    2 /            /      ___\\ /            /      ___\\
5*pi *\pi - 2*I*log\2 + \/ 3 //*\pi - 2*I*log\2 - \/ 3 //
---------------------------------------------------------
                           144
5π2(π−2ilog⁡(2−3))(π−2ilog⁡(3+2))144\frac{5 \pi^{2} \left(\pi - 2 i \log{\left(2 - \sqrt{3} \right)}\right) \left(\pi - 2 i \log{\left(\sqrt{3} + 2 \right)}\right)}{144} 
5*pi^2*(pi - 2*i*log(2 + sqrt(3)))*(pi - 2*i*log(2 - sqrt(3)))/144
Respuesta rápida [src] 
     -5*pi
x1 = -----
       6
x1=−5π6x_{1} = - \frac{5 \pi}{6} 
     -pi 
x2 = ----
      6
x2=−π6x_{2} = - \frac{\pi}{6} 
     pi        /      ___\
x3 = -- - I*log\2 - \/ 3 /
     2
x3=π2−ilog⁡(2−3)x_{3} = \frac{\pi}{2} - i \log{\left(2 - \sqrt{3} \right)} 
     pi        /      ___\
x4 = -- - I*log\2 + \/ 3 /
     2
x4=π2−ilog⁡(3+2)x_{4} = \frac{\pi}{2} - i \log{\left(\sqrt{3} + 2 \right)} 
x4 = pi/2 - i*log(sqrt(3) + 2)
Respuesta numérica [src] 
x1 = -15.1843644923507
x2 = 18.3259571459405
x3 = 66.497044500984
x4 = 93.7241808320955
x5 = -57.0722665402146
x6 = -25.6563400043166
x7 = 56.025068989018
x8 = 47.6474885794452
x9 = -78.0162175641465
x10 = 5.75958653158129
x11 = -82.2050077689329
x12 = -367.042741694407
x13 = 28.7979326579064
x14 = -0.523598775598299
x15 = 97.9129710368819
x16 = 68.5914396033772
x17 = -90.5825881785057
x18 = 43.4586983746588
x19 = 100.007366139275
x20 = -46.6002910282486
x21 = 30.8923277602996
x22 = 37.1755130674792
x23 = -63.3554518473942
x24 = -27.7507351067098
x25 = -34.0339204138894
x26 = -44.5058959258554
x27 = 79.0634151153431
x28 = 91.6297857297023
x29 = -65.4498469497874
x30 = 72.7802298081635
x31 = 60.2138591938044
x32 = 206.821516361328
x33 = -71.733032256967
x34 = -31.9395253114962
x35 = 53.9306738866248
x36 = 3.66519142918809
x37 = 22.5147473507269
x38 = -84.2994028713261
x39 = -88.4881930761125
x40 = -40.317105721069
x41 = -69.6386371545737
x42 = -50.789081233035
x43 = -6.80678408277789
x44 = -19.3731546971371
x45 = 87.4409955249159
x46 = 16.2315620435473
x47 = -38.2227106186758
x48 = 62.3082542961976
x49 = -94.7713783832921
x50 = 49.7418836818384
x51 = 85.3466004225227
x52 = 74.8746249105567
x53 = 12.0427718387609
x54 = -21.4675497995303
x55 = -59.1666616426078
x56 = -13.0899693899575
x57 = 41.3643032722656
x58 = 24.60914245312
x59 = 35.081117965086
x60 = -75.9218224617533
x61 = 9.94837673636768
x62 = -2.61799387799149
x62 = -2.61799387799149
Gráfico
cos(2*x)+3*sin(x)+1=0 la ecuación
    © Sr Examen – Калькулятор