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8tan(θ/2)+8cos(θ)tan(θ/2)=1

Solución

8 tan (\frac{θ}{2}) + 8 cos (θ) tan (\frac{θ}{2}) = 1

Solución

θ = 0.12532 \dots + 2 πn, θ = π - 0.12532 \dots + 2 πn

Grados

θ = 7.18075 \dots^{\circ} + 36 0^{\circ} n, θ = 172.81924 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

8 tan (\frac{θ}{2}) + 8 cos (θ) tan (\frac{θ}{2}) = 1

Restar

1

de ambos lados

8 tan (\frac{θ}{2}) + 8 cos (θ) tan (\frac{θ}{2}) - 1 = 0

Sea:

u = \frac{θ}{2}

8 tan (u) + 8 cos (2 u) tan (u) - 1 = 0

Re-escribir usando identidades trigonométricas

- 1 + 8 tan (u) + 8 cos (2 u) tan (u)

Utilizar la identidad trigonométrica del ángulo doble:

cos (2 x) = 2 cos^{2} (x) - 1

= - 1 + 8 tan (u) + 8 (2 cos^{2} (u) - 1) tan (u)

Simplificar

- 1 + 8 tan (u) + 8 (2 cos^{2} (u) - 1) tan (u) : 16 cos^{2} (u) tan (u) - 1

- 1 + 8 tan (u) + 8 (2 cos^{2} (u) - 1) tan (u)

= - 1 + 8 tan (u) + 8 tan (u) (2 cos^{2} (u) - 1)

Expandir

8 tan (u) (2 cos^{2} (u) - 1) : 16 cos^{2} (u) tan (u) - 8 tan (u)

8 tan (u) (2 cos^{2} (u) - 1)

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = 8 tan (u), b = 2 cos^{2} (u), c = 1

= 8 tan (u) \cdot 2 cos^{2} (u) - 8 tan (u) \cdot 1

= 8 \cdot 2 cos^{2} (u) tan (u) - 8 \cdot 1 \cdot tan (u)

Simplificar

8 \cdot 2 cos^{2} (u) tan (u) - 8 \cdot 1 \cdot tan (u) : 16 cos^{2} (u) tan (u) - 8 tan (u)

8 \cdot 2 cos^{2} (u) tan (u) - 8 \cdot 1 \cdot tan (u)

Multiplicar los numeros:

8 \cdot 2 = 16

= 16 cos^{2} (u) tan (u) - 8 \cdot 1 \cdot tan (u)

Multiplicar los numeros:

8 \cdot 1 = 8

= 16 cos^{2} (u) tan (u) - 8 tan (u)

= 16 cos^{2} (u) tan (u) - 8 tan (u)

= - 1 + 8 tan (u) + 16 cos^{2} (u) tan (u) - 8 tan (u)

Simplificar

- 1 + 8 tan (u) + 16 cos^{2} (u) tan (u) - 8 tan (u) : 16 cos^{2} (u) tan (u) - 1

- 1 + 8 tan (u) + 16 cos^{2} (u) tan (u) - 8 tan (u)

Agrupar términos semejantes

= 8 tan (u) + 16 cos^{2} (u) tan (u) - 8 tan (u) - 1

Sumar elementos similares:

8 tan (u) - 8 tan (u) = 0

= 16 cos^{2} (u) tan (u) - 1

= 16 cos^{2} (u) tan (u) - 1

= 16 cos^{2} (u) tan (u) - 1

Utilizar la identidad trigonométrica básica:

tan (x) = \frac{sin ( x )}{cos ( x )}

= - 1 + 16 cos^{2} (u) \frac{sin ( u )}{cos ( u )}

16 cos^{2} (u) \frac{sin ( u )}{cos ( u )} = 16 sin (u) cos (u)

16 cos^{2} (u) \frac{sin ( u )}{cos ( u )}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{sin ( u ) \cdot 16 cos ^{2} ( u )}{cos ( u )}

Eliminar los terminos comunes:

cos (u)

= 16 sin (u) cos (u)

= - 1 + 16 sin (u) cos (u)

Utilizar la identidad trigonométrica del ángulo doble:

2 sin (x) cos (x) = sin (2 x)

sin (x) cos (x) = \frac{sin ( 2 x )}{2}

= - 1 + 16 \cdot \frac{sin ( 2 u )}{2}

- 1 + 16 \cdot \frac{sin ( 2 u )}{2} = 0

16 \cdot \frac{sin ( 2 u )}{2} = 8 sin (2 u)

16 \cdot \frac{sin ( 2 u )}{2}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{sin ( 2 u ) \cdot 16}{2}

Dividir:

\frac{16}{2} = 8

= 8 sin (2 u)

- 1 + 8 sin (2 u) = 0

Mover

1

al lado derecho

- 1 + 8 sin (2 u) = 0

Sumar

1

a ambos lados

- 1 + 8 sin (2 u) + 1 = 0 + 1

Simplificar

8 sin (2 u) = 1

8 sin (2 u) = 1

Dividir ambos lados entre

8

8 sin (2 u) = 1

Dividir ambos lados entre

8

\frac{8 sin ( 2 u )}{8} = \frac{1}{8}

Simplificar

sin (2 u) = \frac{1}{8}

sin (2 u) = \frac{1}{8}

Aplicar propiedades trigonométricas inversas

sin (2 u) = \frac{1}{8}

Soluciones generales para

sin (2 u) = \frac{1}{8}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

2 u = arcsin (\frac{1}{8}) + 2 πn, 2 u = π - arcsin (\frac{1}{8}) + 2 πn

2 u = arcsin (\frac{1}{8}) + 2 πn, 2 u = π - arcsin (\frac{1}{8}) + 2 πn

Resolver

2 u = arcsin (\frac{1}{8}) + 2 πn : u = \frac{arcsin ( \frac{1}{8} )}{2} + πn

2 u = arcsin (\frac{1}{8}) + 2 πn

Dividir ambos lados entre

2

2 u = arcsin (\frac{1}{8}) + 2 πn

Dividir ambos lados entre

2

\frac{2 u}{2} = \frac{arcsin ( \frac{1}{8} )}{2} + \frac{2 πn}{2}

Simplificar

u = \frac{arcsin ( \frac{1}{8} )}{2} + πn

u = \frac{arcsin ( \frac{1}{8} )}{2} + πn

Resolver

2 u = π - arcsin (\frac{1}{8}) + 2 πn : u = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn

2 u = π - arcsin (\frac{1}{8}) + 2 πn

Dividir ambos lados entre

2

2 u = π - arcsin (\frac{1}{8}) + 2 πn

Dividir ambos lados entre

2

\frac{2 u}{2} = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + \frac{2 πn}{2}

Simplificar

u = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn

u = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn

u = \frac{arcsin ( \frac{1}{8} )}{2} + πn, u = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn

Sustituir en la ecuación

u = \frac{θ}{2}

\frac{θ}{2} = \frac{arcsin ( \frac{1}{8} )}{2} + πn : θ = arcsin (\frac{1}{8}) + 2 πn

\frac{θ}{2} = \frac{arcsin ( \frac{1}{8} )}{2} + πn

Multiplicar ambos lados por

2

\frac{θ}{2} = \frac{arcsin ( \frac{1}{8} )}{2} + πn

Multiplicar ambos lados por

2

\frac{2 θ}{2} = 2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn

Simplificar

\frac{2 θ}{2} = 2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn

Simplificar

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Dividir:

\frac{2}{2} = 1

= θ

Simplificar

2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn : arcsin (\frac{1}{8}) + 2 πn

2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn

2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} = arcsin (\frac{1}{8})

2 \cdot \frac{arcsin ( \frac{1}{8} )}{2}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{arcsin ( \frac{1}{8} ) \cdot 2}{2}

Eliminar los terminos comunes:

2

= arcsin (\frac{1}{8})

= arcsin (\frac{1}{8}) + 2 πn

θ = arcsin (\frac{1}{8}) + 2 πn

θ = arcsin (\frac{1}{8}) + 2 πn

θ = arcsin (\frac{1}{8}) + 2 πn

\frac{θ}{2} = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn : θ = π - arcsin (\frac{1}{8}) + 2 πn

\frac{θ}{2} = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn

Multiplicar ambos lados por

2

\frac{θ}{2} = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn

Multiplicar ambos lados por

2

\frac{2 θ}{2} = 2 \cdot \frac{π}{2} - 2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn

Simplificar

\frac{2 θ}{2} = 2 \cdot \frac{π}{2} - 2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn

Simplificar

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Dividir:

\frac{2}{2} = 1

= θ

Simplificar

2 \cdot \frac{π}{2} - 2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn : π - arcsin (\frac{1}{8}) + 2 πn

2 \cdot \frac{π}{2} - 2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{π 2}{2}

Eliminar los terminos comunes:

2

= π

2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} = arcsin (\frac{1}{8})

2 \cdot \frac{arcsin ( \frac{1}{8} )}{2}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{arcsin ( \frac{1}{8} ) \cdot 2}{2}

Eliminar los terminos comunes:

2

= arcsin (\frac{1}{8})

= π - arcsin (\frac{1}{8}) + 2 πn

θ = π - arcsin (\frac{1}{8}) + 2 πn

θ = π - arcsin (\frac{1}{8}) + 2 πn

θ = π - arcsin (\frac{1}{8}) + 2 πn

θ = arcsin (\frac{1}{8}) + 2 πn, θ = π - arcsin (\frac{1}{8}) + 2 πn

Mostrar soluciones en forma decimal

θ = 0.12532 \dots + 2 πn, θ = π - 0.12532 \dots + 2 πn

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