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sin((π(5x+4))/(15))=(sqrt(3))/(2) la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
                      ___
   /pi*(5*x + 4)\   \/ 3 
sin|------------| = -----
   \     15     /     2
sin⁡(π(5x+4)15)=32\sin{\left(\frac{\pi \left(5 x + 4\right)}{15} \right)} = \frac{\sqrt{3}}{2}
Simplificar
Solución detallada
Tenemos la ecuación
sin⁡(π(5x+4)15)=32\sin{\left(\frac{\pi \left(5 x + 4\right)}{15} \right)} = \frac{\sqrt{3}}{2}
es la ecuación trigonométrica más simple
Esta ecuación se reorganiza en
πx3+4π15=2πn+asin⁡(32)\frac{\pi x}{3} + \frac{4 \pi}{15} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{3}}{2} \right)}
πx3+4π15=2πn−asin⁡(32)+π\frac{\pi x}{3} + \frac{4 \pi}{15} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{3}}{2} \right)} + \pi
O
πx3+4π15=2πn+π3\frac{\pi x}{3} + \frac{4 \pi}{15} = 2 \pi n + \frac{\pi}{3}
πx3+4π15=2πn+2π3\frac{\pi x}{3} + \frac{4 \pi}{15} = 2 \pi n + \frac{2 \pi}{3}
, donde n es cualquier número entero
Transportemos
4π15\frac{4 \pi}{15}
al miembro derecho de la ecuación
con el signo opuesto, en total:
πx3=2πn+π15\frac{\pi x}{3} = 2 \pi n + \frac{\pi}{15}
πx3=2πn+2π5\frac{\pi x}{3} = 2 \pi n + \frac{2 \pi}{5}
Dividamos ambos miembros de la ecuación obtenida en
π3\frac{\pi}{3}
obtenemos la respuesta:
x1=3(2πn+π15)πx_{1} = \frac{3 \left(2 \pi n + \frac{\pi}{15}\right)}{\pi}
x2=3(2πn+2π5)πx_{2} = \frac{3 \left(2 \pi n + \frac{2 \pi}{5}\right)}{\pi}
Gráfica
0-80-60-40-2020406080-1001002-2
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Respuesta rápida [src] 
x1 = 1/5
x1=15x_{1} = \frac{1}{5} 
x2 = 6/5
x2=65x_{2} = \frac{6}{5} 
x2 = 6/5
Suma y producto de raíces [src] 
suma
1/5 + 6/5
15+65\frac{1}{5} + \frac{6}{5} 
=
7/5
75\frac{7}{5} 
producto
 6 
---
5*5
65⋅5\frac{6}{5 \cdot 5} 
=
6/25
625\frac{6}{25} 
6/25
Respuesta numérica [src] 
x1 = -29.8
x2 = -160.8
x3 = -106.8
x4 = -41.8
x5 = -256.8
x6 = 96.2
x7 = 55.2
x8 = -244.8
x9 = -178.8
x10 = 84.2
x11 = 97.2
x12 = 73.2
x13 = -70.8
x14 = -226.8
x15 = 72.2
x16 = -100.8
x17 = -166.8
x18 = -83.8
x19 = -11.8
x20 = -220.8
x21 = 91.2
x22 = 1.2
x23 = -94.8
x24 = 12.2
x25 = 61.2
x26 = -52.8
x27 = -136.8
x28 = 48.2
x29 = 18.2
x30 = 6.2
x31 = -148.8
x32 = -17.8
x33 = -76.8
x34 = -35.8
x35 = 42.2
x36 = -46.8
x37 = -82.8
x38 = -34.8
x39 = 0.2
x40 = 36.2
x41 = 25.2
x42 = -4.8
x43 = -250.8
x44 = -112.8
x45 = 43.2
x46 = 31.2
x47 = 78.2
x48 = 67.2
x49 = 24.2
x50 = -124.8
x51 = 7.2
x52 = 85.2
x53 = -10.8
x54 = -202.8
x55 = -89.8
x56 = 54.2
x57 = -22.8
x58 = -71.8
x59 = -95.8
x60 = -232.8
x61 = 66.2
x62 = -208.8
x63 = 37.2
x64 = -47.8
x65 = -184.8
x66 = 79.2
x67 = -154.8
x68 = -65.8
x69 = -190.8
x70 = -16.8
x71 = 19.2
x72 = -196.8
x73 = -58.8
x74 = 13.2
x75 = -88.8
x76 = -28.8
x77 = -23.8
x78 = 49.2
x79 = -172.8
x80 = -40.8
x81 = -130.8
x82 = 30.2
x83 = -59.8
x84 = -118.8
x85 = -214.8
x86 = 60.2
x87 = -53.8
x88 = -142.8
x89 = -262.8
x90 = -77.8
x91 = -64.8
x92 = -238.8
x93 = -5.8
x94 = 90.2
x94 = 90.2
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