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Popular Trigonometría >

(-2+sqrt(3))/2 =-1+cos(4θ+135)

Solución

\frac{- 2 + 3}{2} = - 1 + cos (4 θ + 135)

Solución

θ = \frac{π}{24} - \frac{135}{4} + \frac{πn}{2}, θ = \frac{π}{2} - \frac{π}{24} - \frac{135}{4} + \frac{πn}{2}

Grados

θ = - 1926.23255 \dots^{\circ} + 9 0^{\circ} n, θ = - 1851.23255 \dots^{\circ} + 9 0^{\circ} n

Pasos de solución

\frac{- 2 + 3}{2} = - 1 + cos (4 θ + 135)

Intercambiar lados

- 1 + cos (4 θ + 135) = \frac{- 2 + 3}{2}

Desplace

1

a la derecha

- 1 + cos (4 θ + 135) = \frac{- 2 + 3}{2}

Sumar

1

a ambos lados

- 1 + cos (4 θ + 135) + 1 = \frac{- 2 + 3}{2} + 1

Simplificar

cos (4 θ + 135) = \frac{- 2 + 3}{2} + 1

cos (4 θ + 135) = \frac{- 2 + 3}{2} + 1

Aplicar propiedades trigonométricas inversas

cos (4 θ + 135) = \frac{- 2 + 3}{2} + 1

Soluciones generales para

cos (4 θ + 135) = \frac{- 2 + 3}{2} + 1

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

4 θ + 135 = arccos (\frac{- 2 + 3}{2} + 1) + 2 πn, 4 θ + 135 = 2 π - arccos (\frac{- 2 + 3}{2} + 1) + 2 πn

4 θ + 135 = arccos (\frac{- 2 + 3}{2} + 1) + 2 πn, 4 θ + 135 = 2 π - arccos (\frac{- 2 + 3}{2} + 1) + 2 πn

Resolver

4 θ + 135 = arccos (\frac{- 2 + 3}{2} + 1) + 2 πn : θ = \frac{π}{24} - \frac{135}{4} + \frac{πn}{2}

4 θ + 135 = arccos (\frac{- 2 + 3}{2} + 1) + 2 πn

Simplificar

arccos (\frac{- 2 + 3}{2} + 1) + 2 πn : \frac{π}{6} + 2 πn

arccos (\frac{- 2 + 3}{2} + 1) + 2 πn

Utilizar la siguiente identidad trivial:

arccos (\frac{3 - 2}{2} + 1) = \frac{π}{6}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= \frac{π}{6} + 2 πn

4 θ + 135 = \frac{π}{6} + 2 πn

Desplace

135

a la derecha

4 θ + 135 = \frac{π}{6} + 2 πn

Restar

135

de ambos lados

4 θ + 135 - 135 = \frac{π}{6} + 2 πn - 135

Simplificar

4 θ = \frac{π}{6} + 2 πn - 135

4 θ = \frac{π}{6} + 2 πn - 135

Dividir ambos lados entre

4

4 θ = \frac{π}{6} + 2 πn - 135

Dividir ambos lados entre

4

\frac{4 θ}{4} = \frac{\frac{π}{6}}{4} + \frac{2 πn}{4} - \frac{135}{4}

Simplificar

\frac{4 θ}{4} = \frac{\frac{π}{6}}{4} + \frac{2 πn}{4} - \frac{135}{4}

Simplificar

\frac{4 θ}{4} : θ

\frac{4 θ}{4}

Dividir:

\frac{4}{4} = 1

= θ

Simplificar

\frac{\frac{π}{6}}{4} + \frac{2 πn}{4} - \frac{135}{4} : \frac{π}{24} - \frac{135}{4} + \frac{πn}{2}

\frac{\frac{π}{6}}{4} + \frac{2 πn}{4} - \frac{135}{4}

Agrupar términos semejantes

= - \frac{135}{4} + \frac{2 πn}{4} + \frac{\frac{π}{6}}{4}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Eliminar los terminos comunes:

2

= \frac{πn}{2}

\frac{\frac{π}{6}}{4} = \frac{π}{24}

\frac{\frac{π}{6}}{4}

Aplicar las propiedades de las fracciones:

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

= \frac{π}{6 \cdot 4}

Multiplicar los numeros:

6 \cdot 4 = 24

= \frac{π}{24}

= - \frac{135}{4} + \frac{πn}{2} + \frac{π}{24}

Agrupar términos semejantes

= \frac{π}{24} - \frac{135}{4} + \frac{πn}{2}

θ = \frac{π}{24} - \frac{135}{4} + \frac{πn}{2}

θ = \frac{π}{24} - \frac{135}{4} + \frac{πn}{2}

θ = \frac{π}{24} - \frac{135}{4} + \frac{πn}{2}

Resolver

4 θ + 135 = 2 π - arccos (\frac{- 2 + 3}{2} + 1) + 2 πn : θ = \frac{π}{2} - \frac{π}{24} - \frac{135}{4} + \frac{πn}{2}

4 θ + 135 = 2 π - arccos (\frac{- 2 + 3}{2} + 1) + 2 πn

Simplificar

2 π - arccos (\frac{- 2 + 3}{2} + 1) + 2 πn : 2 π - \frac{π}{6} + 2 πn

2 π - arccos (\frac{- 2 + 3}{2} + 1) + 2 πn

Utilizar la siguiente identidad trivial:

arccos (\frac{3 - 2}{2} + 1) = \frac{π}{6}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= 2 π - \frac{π}{6} + 2 πn

4 θ + 135 = 2 π - \frac{π}{6} + 2 πn

Desplace

135

a la derecha

4 θ + 135 = 2 π - \frac{π}{6} + 2 πn

Restar

135

de ambos lados

4 θ + 135 - 135 = 2 π - \frac{π}{6} + 2 πn - 135

Simplificar

4 θ = 2 π - \frac{π}{6} + 2 πn - 135

4 θ = 2 π - \frac{π}{6} + 2 πn - 135

Dividir ambos lados entre

4

4 θ = 2 π - \frac{π}{6} + 2 πn - 135

Dividir ambos lados entre

4

\frac{4 θ}{4} = \frac{2 π}{4} - \frac{\frac{π}{6}}{4} + \frac{2 πn}{4} - \frac{135}{4}

Simplificar

\frac{4 θ}{4} = \frac{2 π}{4} - \frac{\frac{π}{6}}{4} + \frac{2 πn}{4} - \frac{135}{4}

Simplificar

\frac{4 θ}{4} : θ

\frac{4 θ}{4}

Dividir:

\frac{4}{4} = 1

= θ

Simplificar

\frac{2 π}{4} - \frac{\frac{π}{6}}{4} + \frac{2 πn}{4} - \frac{135}{4} : \frac{π}{2} - \frac{π}{24} - \frac{135}{4} + \frac{πn}{2}

\frac{2 π}{4} - \frac{\frac{π}{6}}{4} + \frac{2 πn}{4} - \frac{135}{4}

Agrupar términos semejantes

= \frac{2 π}{4} - \frac{135}{4} + \frac{2 πn}{4} - \frac{\frac{π}{6}}{4}

\frac{2 π}{4} = \frac{π}{2}

\frac{2 π}{4}

Eliminar los terminos comunes:

2

= \frac{π}{2}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Eliminar los terminos comunes:

2

= \frac{πn}{2}

\frac{\frac{π}{6}}{4} = \frac{π}{24}

\frac{\frac{π}{6}}{4}

Aplicar las propiedades de las fracciones:

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

= \frac{π}{6 \cdot 4}

Multiplicar los numeros:

6 \cdot 4 = 24

= \frac{π}{24}

= \frac{π}{2} - \frac{135}{4} + \frac{πn}{2} - \frac{π}{24}

Agrupar términos semejantes

= \frac{π}{2} - \frac{π}{24} - \frac{135}{4} + \frac{πn}{2}

θ = \frac{π}{2} - \frac{π}{24} - \frac{135}{4} + \frac{πn}{2}

θ = \frac{π}{2} - \frac{π}{24} - \frac{135}{4} + \frac{πn}{2}

θ = \frac{π}{2} - \frac{π}{24} - \frac{135}{4} + \frac{πn}{2}

θ = \frac{π}{24} - \frac{135}{4} + \frac{πn}{2}, θ = \frac{π}{2} - \frac{π}{24} - \frac{135}{4} + \frac{πn}{2}

Gráfica

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