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4cos^2x-sinx+1=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
     2                    
4*cos (x) - sin(x) + 1 = 0
(−sin⁡(x)+4cos⁡2(x))+1=0\left(- \sin{\left(x \right)} + 4 \cos^{2}{\left(x \right)}\right) + 1 = 0
Simplificar
Solución detallada
Tenemos la ecuación
(−sin⁡(x)+4cos⁡2(x))+1=0\left(- \sin{\left(x \right)} + 4 \cos^{2}{\left(x \right)}\right) + 1 = 0
cambiamos
−4sin⁡2(x)−sin⁡(x)+5=0- 4 \sin^{2}{\left(x \right)} - \sin{\left(x \right)} + 5 = 0
−4sin⁡2(x)−sin⁡(x)+5=0- 4 \sin^{2}{\left(x \right)} - \sin{\left(x \right)} + 5 = 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=−4a = -4
b=−1b = -1
c=5c = 5
, entonces
D = b^2 - 4 * a * c = 

(-1)^2 - 4 * (-4) * (5) = 81

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

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

o
w1=−54w_{1} = - \frac{5}{4}
w2=1w_{2} = 1
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⁡(−54)x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{5}{4} \right)}
x1=2πn−asin⁡(54)x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{5}{4} \right)}
x2=2πn+asin⁡(w2)x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}
x2=2πn+asin⁡(1)x_{2} = 2 \pi n + \operatorname{asin}{\left(1 \right)}
x2=2πn+π2x_{2} = 2 \pi n + \frac{\pi}{2}
x3=2πn−asin⁡(w1)+πx_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi
x3=2πn+π−asin⁡(−54)x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(- \frac{5}{4} \right)}
x3=2πn+π+asin⁡(54)x_{3} = 2 \pi n + \pi + \operatorname{asin}{\left(\frac{5}{4} \right)}
x4=2πn−asin⁡(w2)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi
x4=2πn−asin⁡(1)+πx_{4} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi
x4=2πn+π2x_{4} = 2 \pi n + \frac{\pi}{2}
Gráfica
0-80-60-40-2020406080-100100010
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
     pi
x1 = --
     2
x1=π2x_{1} = \frac{\pi}{2} 
           /    /4   3*I\\         /    /4   3*I\\
x2 = - 2*re|atan|- - ---|| - 2*I*im|atan|- - ---||
           \    \5    5 //         \    \5    5 //
x2=−2re⁡(atan⁡(45−3i5))−2iim⁡(atan⁡(45−3i5))x_{2} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{4}{5} - \frac{3 i}{5} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{4}{5} - \frac{3 i}{5} \right)}\right)} 
           /    /4   3*I\\         /    /4   3*I\\
x3 = - 2*re|atan|- + ---|| - 2*I*im|atan|- + ---||
           \    \5    5 //         \    \5    5 //
x3=−2re⁡(atan⁡(45+3i5))−2iim⁡(atan⁡(45+3i5))x_{3} = - 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{4}{5} + \frac{3 i}{5} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{4}{5} + \frac{3 i}{5} \right)}\right)} 
x3 = -2*re(atan(4/5 + 3*i/5)) - 2*i*im(atan(4/5 + 3*i/5))
Suma y producto de raíces [src] 
suma
pi         /    /4   3*I\\         /    /4   3*I\\         /    /4   3*I\\         /    /4   3*I\\
-- + - 2*re|atan|- - ---|| - 2*I*im|atan|- - ---|| + - 2*re|atan|- + ---|| - 2*I*im|atan|- + ---||
2          \    \5    5 //         \    \5    5 //         \    \5    5 //         \    \5    5 //
(−2re⁡(atan⁡(45+3i5))−2iim⁡(atan⁡(45+3i5)))+(π2+(−2re⁡(atan⁡(45−3i5))−2iim⁡(atan⁡(45−3i5))))\left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{4}{5} + \frac{3 i}{5} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{4}{5} + \frac{3 i}{5} \right)}\right)}\right) + \left(\frac{\pi}{2} + \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{4}{5} - \frac{3 i}{5} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{4}{5} - \frac{3 i}{5} \right)}\right)}\right)\right) 
=
pi       /    /4   3*I\\       /    /4   3*I\\         /    /4   3*I\\         /    /4   3*I\\
-- - 2*re|atan|- - ---|| - 2*re|atan|- + ---|| - 2*I*im|atan|- - ---|| - 2*I*im|atan|- + ---||
2        \    \5    5 //       \    \5    5 //         \    \5    5 //         \    \5    5 //
−2re⁡(atan⁡(45−3i5))−2re⁡(atan⁡(45+3i5))+π2−2iim⁡(atan⁡(45+3i5))−2iim⁡(atan⁡(45−3i5))- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{4}{5} - \frac{3 i}{5} \right)}\right)} - 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{4}{5} + \frac{3 i}{5} \right)}\right)} + \frac{\pi}{2} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{4}{5} + \frac{3 i}{5} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{4}{5} - \frac{3 i}{5} \right)}\right)} 
producto
pi /      /    /4   3*I\\         /    /4   3*I\\\ /      /    /4   3*I\\         /    /4   3*I\\\
--*|- 2*re|atan|- - ---|| - 2*I*im|atan|- - ---|||*|- 2*re|atan|- + ---|| - 2*I*im|atan|- + ---|||
2  \      \    \5    5 //         \    \5    5 /// \      \    \5    5 //         \    \5    5 ///
π2(−2re⁡(atan⁡(45−3i5))−2iim⁡(atan⁡(45−3i5)))(−2re⁡(atan⁡(45+3i5))−2iim⁡(atan⁡(45+3i5)))\frac{\pi}{2} \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{4}{5} - \frac{3 i}{5} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{4}{5} - \frac{3 i}{5} \right)}\right)}\right) \left(- 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{4}{5} + \frac{3 i}{5} \right)}\right)} - 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{4}{5} + \frac{3 i}{5} \right)}\right)}\right) 
=
     /    /    /4   3*I\\     /    /4   3*I\\\ /    /    /4   3*I\\     /    /4   3*I\\\
2*pi*|I*im|atan|- - ---|| + re|atan|- - ---|||*|I*im|atan|- + ---|| + re|atan|- + ---|||
     \    \    \5    5 //     \    \5    5 /// \    \    \5    5 //     \    \5    5 ///
2π(re⁡(atan⁡(45−3i5))+iim⁡(atan⁡(45−3i5)))(re⁡(atan⁡(45+3i5))+iim⁡(atan⁡(45+3i5)))2 \pi \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{4}{5} - \frac{3 i}{5} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{4}{5} - \frac{3 i}{5} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{4}{5} + \frac{3 i}{5} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{4}{5} + \frac{3 i}{5} \right)}\right)}\right) 
2*pi*(i*im(atan(4/5 - 3*i/5)) + re(atan(4/5 - 3*i/5)))*(i*im(atan(4/5 + 3*i/5)) + re(atan(4/5 + 3*i/5)))
Respuesta numérica [src] 
x1 = -10.9955745507927
x2 = 1.5707965487099
x3 = 14.1371671054705
x4 = -23.5619452560558
x5 = -86.3937977628397
x6 = -80.1106125794063
x7 = 58.1194639689786
x8 = -61.2610569677124
x9 = 102.101761119382
x10 = 11015.994639417
x11 = 83.2522055898627
x12 = -36.1283154190274
x13 = 32.986723027975
x14 = -4.71238911743622
x15 = 58.1194644304554
x16 = -73.8274272800187
x17 = 39.2699080118312
x18 = -54.9778712408359
x19 = 76.9690201096197
x20 = -48.6946858609913
x21 = -29.8451301660968
x22 = -400.5530636364
x23 = -42.411500606946
x24 = 7.8539816297258
x25 = 7.8539817408679
x26 = -17.2787598123494
x27 = 70.6858344990791
x28 = -92.6769830121323
x29 = -73.8274270126378
x30 = 64.4026493085237
x31 = 83.2522051441511
x32 = 76.9690197502489
x33 = -92.6769833644259
x34 = -10.9955741052319
x35 = -98.9601683784854
x36 = 14.1371668519517
x37 = 26.7035373432502
x38 = 76.9690201583539
x39 = 51.8362789003523
x40 = -54.9778717005513
x41 = -17.2787599640228
x42 = 32.9867225994861
x43 = -48.6946862430865
x44 = -17.278759767136
x45 = 39.269908440032
x46 = -23.561945009324
x47 = 89.53539085929
x48 = 45.5530937041236
x49 = -29.8451300962681
x50 = -98.9601688497483
x51 = 95.8185760594783
x52 = -67.544242167931
x53 = -4.71238871026182
x54 = 20.4203521494903
x54 = 20.4203521494903
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