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sin^2x+15cosx+15=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   2                        
sin (x) + 15*cos(x) + 15 = 0
(sin⁡2(x)+15cos⁡(x))+15=0\left(\sin^{2}{\left(x \right)} + 15 \cos{\left(x \right)}\right) + 15 = 0
Simplificar
Solución detallada
Tenemos la ecuación
(sin⁡2(x)+15cos⁡(x))+15=0\left(\sin^{2}{\left(x \right)} + 15 \cos{\left(x \right)}\right) + 15 = 0
cambiamos
sin⁡2(x)+15cos⁡(x)+15=0\sin^{2}{\left(x \right)} + 15 \cos{\left(x \right)} + 15 = 0
−cos⁡2(x)+15cos⁡(x)+16=0- \cos^{2}{\left(x \right)} + 15 \cos{\left(x \right)} + 16 = 0
Sustituimos
w=cos⁡(x)w = \cos{\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=−1a = -1
b=15b = 15
c=16c = 16
, entonces
D = b^2 - 4 * a * c = 

(15)^2 - 4 * (-1) * (16) = 289

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

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

o
w1=−1w_{1} = -1
w2=16w_{2} = 16
hacemos cambio inverso
cos⁡(x)=w\cos{\left(x \right)} = w
Tenemos la ecuación
cos⁡(x)=w\cos{\left(x \right)} = w
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
x=πn+acos⁡(w)x = \pi n + \operatorname{acos}{\left(w \right)}
x=πn+acos⁡(w)−πx = \pi n + \operatorname{acos}{\left(w \right)} - \pi
O
x=πn+acos⁡(w)x = \pi n + \operatorname{acos}{\left(w \right)}
x=πn+acos⁡(w)−πx = \pi n + \operatorname{acos}{\left(w \right)} - \pi
, donde n es cualquier número entero
sustituimos w:
x1=πn+acos⁡(w1)x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}
x1=πn+acos⁡(−1)x_{1} = \pi n + \operatorname{acos}{\left(-1 \right)}
x1=πn+πx_{1} = \pi n + \pi
x2=πn+acos⁡(w2)x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}
x2=πn+acos⁡(16)x_{2} = \pi n + \operatorname{acos}{\left(16 \right)}
x2=πn+acos⁡(16)x_{2} = \pi n + \operatorname{acos}{\left(16 \right)}
x3=πn+acos⁡(w1)−πx_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi
x3=πn−π+acos⁡(−1)x_{3} = \pi n - \pi + \operatorname{acos}{\left(-1 \right)}
x3=πnx_{3} = \pi n
x4=πn+acos⁡(w2)−πx_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi
x4=πn−π+acos⁡(16)x_{4} = \pi n - \pi + \operatorname{acos}{\left(16 \right)}
x4=πn−π+acos⁡(16)x_{4} = \pi n - \pi + \operatorname{acos}{\left(16 \right)}
Gráfica
0-80-60-40-2020406080-100100050
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
               /  _____\
               |\/ 255 |
x1 = -2*I*atanh|-------|
               \   17  /
x1=−2iatanh⁡(25517)x_{1} = - 2 i \operatorname{atanh}{\left(\frac{\sqrt{255}}{17} \right)} 
              /  _____\
              |\/ 255 |
x2 = 2*I*atanh|-------|
              \   17  /
x2=2iatanh⁡(25517)x_{2} = 2 i \operatorname{atanh}{\left(\frac{\sqrt{255}}{17} \right)} 
x2 = 2*i*atanh(sqrt(255)/17)
Suma y producto de raíces [src] 
suma
           /  _____\            /  _____\
           |\/ 255 |            |\/ 255 |
- 2*I*atanh|-------| + 2*I*atanh|-------|
           \   17  /            \   17  /
−2iatanh⁡(25517)+2iatanh⁡(25517)- 2 i \operatorname{atanh}{\left(\frac{\sqrt{255}}{17} \right)} + 2 i \operatorname{atanh}{\left(\frac{\sqrt{255}}{17} \right)} 
=
0
00 
producto
          /  _____\          /  _____\
          |\/ 255 |          |\/ 255 |
-2*I*atanh|-------|*2*I*atanh|-------|
          \   17  /          \   17  /
−2iatanh⁡(25517)2iatanh⁡(25517)- 2 i \operatorname{atanh}{\left(\frac{\sqrt{255}}{17} \right)} 2 i \operatorname{atanh}{\left(\frac{\sqrt{255}}{17} \right)} 
=
        /  _____\
       2|\/ 255 |
4*atanh |-------|
        \   17  /
4atanh⁡2(25517)4 \operatorname{atanh}^{2}{\left(\frac{\sqrt{255}}{17} \right)} 
4*atanh(sqrt(255)/17)^2
Respuesta numérica [src] 
x1 = -59.6902604645339
x2 = 78.5398161813235
x3 = 84.8230013683715
x4 = 72.2566310277178
x5 = -15.7079632965905
x6 = 97.389372574747
x7 = -21.9911485864428
x8 = 9.42477826134354
x9 = -21.9911489515442
x10 = 72.2566308907209
x11 = -72.2566308667464
x12 = -65.9734460923405
x13 = -53.4070752937247
x14 = 3.14159226996045
x15 = 28.2743337696744
x16 = -34.5575188955148
x17 = 47.1238894250738
x18 = -21.9911484096383
x19 = -28.2743337080309
x20 = -3.14159294551882
x21 = -84.8230012654508
x22 = -47.1238901023175
x23 = -9.4247781354684
x24 = 59.6902606089937
x25 = -78.5398160524763
x26 = 34.5575191475629
x27 = -97.3893724518906
x28 = -15.7079633488247
x29 = 72.2566314176769
x30 = 15.7079634505462
x31 = 40.840704211378
x32 = -91.1061872590165
x33 = 53.4070754180978
x34 = -40.8407041102953
x35 = 21.9911487134469
x36 = 65.9734457529715
x37 = 21.9911485852048
x38 = -40.8407048850241
x39 = -59.6902604577826
x40 = 21.9911481899934
x41 = 65.9734458341826
x42 = 91.1061865803137
x43 = 28.2743338651821
x44 = 34.5575190228802
x45 = -65.9734457649381
x46 = -65.9734455348684
x46 = -65.9734455348684
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