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sin^2*(7.5C)/sin^2*(7C) la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   2/15*c\    
sin |----|    
    \ 2  /    
---------- = 0
   2          
sin (7*c)
sin⁡2(15c2)sin⁡2(7c)=0\frac{\sin^{2}{\left(\frac{15 c}{2} \right)}}{\sin^{2}{\left(7 c \right)}} = 0
Simplificar
Solución detallada
Tenemos la ecuación
sin⁡2(15c2)sin⁡2(7c)=0\frac{\sin^{2}{\left(\frac{15 c}{2} \right)}}{\sin^{2}{\left(7 c \right)}} = 0
cambiamos
1−cos⁡(15c)1−cos⁡(14c)=0\frac{1 - \cos{\left(15 c \right)}}{1 - \cos{\left(14 c \right)}} = 0
sin⁡2(15c2)sin⁡2(7c)=0\frac{\sin^{2}{\left(\frac{15 c}{2} \right)}}{\sin^{2}{\left(7 c \right)}} = 0
Sustituimos
w=sin⁡(15c2)w = \sin{\left(\frac{15 c}{2} \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=1sin⁡2(7c)a = \frac{1}{\sin^{2}{\left(7 c \right)}}
b=0b = 0
c=0c = 0
, entonces
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (sin(7*c)^(-2)) * (0) = 0

Como D = 0 hay sólo una raíz.
w = -b/2a = -0/2/(sin(7*c)^(-2))

w1=0w_{1} = 0
hacemos cambio inverso
sin⁡(15c2)=w\sin{\left(\frac{15 c}{2} \right)} = w
Tenemos la ecuación
sin⁡(15c2)=w\sin{\left(\frac{15 c}{2} \right)} = w
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
15c2=2πn+asin⁡(w)\frac{15 c}{2} = 2 \pi n + \operatorname{asin}{\left(w \right)}
15c2=2πn−asin⁡(w)+π\frac{15 c}{2} = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
O
15c2=2πn+asin⁡(w)\frac{15 c}{2} = 2 \pi n + \operatorname{asin}{\left(w \right)}
15c2=2πn−asin⁡(w)+π\frac{15 c}{2} = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi
, donde n es cualquier número entero
Dividamos ambos miembros de la ecuación obtenida en
152\frac{15}{2}
sustituimos w:
c1=4πn15+2asin⁡(w1)15c_{1} = \frac{4 \pi n}{15} + \frac{2 \operatorname{asin}{\left(w_{1} \right)}}{15}
c1=4πn15+2asin⁡(0)15c_{1} = \frac{4 \pi n}{15} + \frac{2 \operatorname{asin}{\left(0 \right)}}{15}
c1=4πn15c_{1} = \frac{4 \pi n}{15}
c2=4πn15−2asin⁡(w1)15+2π15c_{2} = \frac{4 \pi n}{15} - \frac{2 \operatorname{asin}{\left(w_{1} \right)}}{15} + \frac{2 \pi}{15}
c2=4πn15−2asin⁡(0)15+2π15c_{2} = \frac{4 \pi n}{15} - \frac{2 \operatorname{asin}{\left(0 \right)}}{15} + \frac{2 \pi}{15}
c2=4πn15+2π15c_{2} = \frac{4 \pi n}{15} + \frac{2 \pi}{15}
Gráfica
0-80-60-40-2020406080-100100020000000
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
     2*pi
c1 = ----
      15
c1=2π15c_{1} = \frac{2 \pi}{15} 
c1 = 2*pi/15
Suma y producto de raíces [src] 
suma
2*pi
----
 15
2π15\frac{2 \pi}{15} 
=
2*pi
----
 15
2π15\frac{2 \pi}{15} 
producto
2*pi
----
 15
2π15\frac{2 \pi}{15} 
=
2*pi
----
 15
2π15\frac{2 \pi}{15} 
2*pi/15
Respuesta numérica [src] 
c1 = -3.76991116753428
c2 = 40.2123860044362
c3 = 4.18879023890196
c4 = -57.8053048067812
c5 = 64.0884900879688
c6 = -41.8879021457996
c7 = -55.7109097271288
c8 = -80.0058929052583
c9 = 90.0589893461342
c10 = 54.0353935769418
c11 = 63.6696111038366
c12 = -73.7227076186611
c13 = -33.5103216851037
c14 = -78.3303768383633
c15 = -99.6932068783928
c16 = -39.7935069194202
c17 = -90.058989351523
c18 = 58.2241838850947
c19 = -72.0471914798199
c20 = -47.7522083329067
c21 = -20.1061929473143
c22 = -65.7640062048304
c23 = 92.1533845556184
c24 = -46.0766921954986
c25 = -69.9527964496298
c26 = 11.7286126348605
c27 = -11.7286125779928
c28 = 32.253684574718
c29 = -61.9940950402693
c30 = 2.09439502262831
c31 = 22.2005881230556
c32 = 84.194683197118
c33 = 14.2418867033413
c34 = 80.0058928410346
c35 = -21.7817090414627
c36 = -2.09439506456015
c37 = -54.0353936207766
c38 = 36.0235956549122
c39 = -18.0117978920925
c40 = 66.1828852911588
c41 = 72.0471914643502
c42 = 59.8996999244811
c43 = 30.1592894966635
c44 = 70.3716754173925
c45 = -29.7404104601353
c46 = -36.0235957585474
c47 = -42.3067811485979
c48 = 48.1710873969368
c49 = 28.0648942832288
c50 = -67.8584013630606
c51 = -10.0530965334233
c52 = 74.141586648457
c53 = 76.2359817659643
c54 = -91.7345054956576
c55 = -82.5191669932907
c56 = 26.8082573122071
c57 = -83.7758040788188
c58 = 20.1061929368606
c59 = -13.8230076550098
c60 = -28.0648943438799
c61 = 46.0766921846909
c62 = -64.0884901011257
c63 = 61.9940950345747
c64 = 80.0058929030126
c65 = 98.0176907677684
c66 = -23.8761042129715
c67 = 96.3421746740399
c68 = 23.8761042018758
c69 = 77.0737397341008
c70 = -98.0176907733313
c70 = -98.0176907733313
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