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solvefor x,cos(x/2)=-cos(2x-30)

Solución

resolver para

x, cos (\frac{x}{2}) = - cos (2 x - 30)

Solución

x = \frac{2 π}{3} + 20 + \frac{8 πn}{3}, x = 2 π + 20 + \frac{8 πn}{3}, x = \frac{2 π}{5} + 12 + \frac{8 πn}{5}, x = \frac{6 π}{5} + 12 + \frac{8 πn}{5}

Grados

x = 1265.91559 \dots^{\circ} + 48 0^{\circ} n, x = 1505.91559 \dots^{\circ} + 48 0^{\circ} n, x = 759.54935 \dots^{\circ} + 28 8^{\circ} n, x = 903.54935 \dots^{\circ} + 28 8^{\circ} n

Pasos de solución

cos (\frac{x}{2}) = - cos (2 x - 30)

Restar

- cos (2 x - 30)

de ambos lados

cos (\frac{x}{2}) + cos (2 x - 30) = 0

Re-escribir usando identidades trigonométricas

cos (- 30 + 2 x) + cos (\frac{x}{2})

Utilizar la identidad suma-producto:

cos (s) + cos (t) = 2 cos (\frac{s + t}{2}) cos (\frac{s - t}{2})

= 2 cos (\frac{- 30 + 2 x + \frac{x}{2}}{2}) cos (\frac{- 30 + 2 x - \frac{x}{2}}{2})

Simplificar

2 cos (\frac{- 30 + 2 x + \frac{x}{2}}{2}) cos (\frac{- 30 + 2 x - \frac{x}{2}}{2}) : 2 cos (\frac{- 60 + 5 x}{4}) cos (\frac{- 60 + 3 x}{4})

2 cos (\frac{- 30 + 2 x + \frac{x}{2}}{2}) cos (\frac{- 30 + 2 x - \frac{x}{2}}{2})

\frac{- 30 + 2 x + \frac{x}{2}}{2} = \frac{- 60 + 5 x}{4}

\frac{- 30 + 2 x + \frac{x}{2}}{2}

Simplificar

- 30 + 2 x + \frac{x}{2}

en una fracción:

\frac{- 60 + 5 x}{2}

- 30 + 2 x + \frac{x}{2}

Convertir a fracción:

30 = \frac{30 \cdot 2}{2}, 2 x = \frac{2 x 2}{2}

= - \frac{30 \cdot 2}{2} + \frac{2 x \cdot 2}{2} + \frac{x}{2}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{- 30 \cdot 2 + 2 x \cdot 2 + x}{2}

- 30 \cdot 2 + 2 x \cdot 2 + x = - 60 + 5 x

- 30 \cdot 2 + 2 x \cdot 2 + x

Multiplicar los numeros:

30 \cdot 2 = 60

= - 60 + 2 \cdot 2 x + x

Multiplicar los numeros:

2 \cdot 2 = 4

= - 60 + 4 x + x

Sumar elementos similares:

4 x + x = 5 x

= - 60 + 5 x

= \frac{- 60 + 5 x}{2}

= \frac{\frac{- 60 + 5 x}{2}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{- 60 + 5 x}{2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{- 60 + 5 x}{4}

= 2 cos (\frac{5 x - 60}{4}) cos (\frac{2 x - \frac{x}{2} - 30}{2})

\frac{- 30 + 2 x - \frac{x}{2}}{2} = \frac{- 60 + 3 x}{4}

\frac{- 30 + 2 x - \frac{x}{2}}{2}

Simplificar

- 30 + 2 x - \frac{x}{2}

en una fracción:

\frac{- 60 + 3 x}{2}

- 30 + 2 x - \frac{x}{2}

Convertir a fracción:

30 = \frac{30 \cdot 2}{2}, 2 x = \frac{2 x 2}{2}

= - \frac{30 \cdot 2}{2} + \frac{2 x \cdot 2}{2} - \frac{x}{2}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{- 30 \cdot 2 + 2 x \cdot 2 - x}{2}

- 30 \cdot 2 + 2 x \cdot 2 - x = - 60 + 3 x

- 30 \cdot 2 + 2 x \cdot 2 - x

Multiplicar los numeros:

30 \cdot 2 = 60

= - 60 + 2 \cdot 2 x - x

Multiplicar los numeros:

2 \cdot 2 = 4

= - 60 + 4 x - x

Sumar elementos similares:

4 x - x = 3 x

= - 60 + 3 x

= \frac{- 60 + 3 x}{2}

= \frac{\frac{- 60 + 3 x}{2}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{- 60 + 3 x}{2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{- 60 + 3 x}{4}

= 2 cos (\frac{5 x - 60}{4}) cos (\frac{3 x - 60}{4})

= 2 cos (\frac{- 60 + 5 x}{4}) cos (\frac{- 60 + 3 x}{4})

2 cos (\frac{- 60 + 3 x}{4}) cos (\frac{- 60 + 5 x}{4}) = 0

Resolver cada parte por separado

cos (\frac{- 60 + 3 x}{4}) = 0 or cos (\frac{- 60 + 5 x}{4}) = 0

cos (\frac{- 60 + 3 x}{4}) = 0 : x = \frac{2 π}{3} + 20 + \frac{8 πn}{3}, x = 2 π + 20 + \frac{8 πn}{3}

cos (\frac{- 60 + 3 x}{4}) = 0

Soluciones generales para

cos (\frac{- 60 + 3 x}{4}) = 0

cos (x)

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

\frac{- 60 + 3 x}{4} = \frac{π}{2} + 2 πn, \frac{- 60 + 3 x}{4} = \frac{3 π}{2} + 2 πn

\frac{- 60 + 3 x}{4} = \frac{π}{2} + 2 πn, \frac{- 60 + 3 x}{4} = \frac{3 π}{2} + 2 πn

Resolver

\frac{- 60 + 3 x}{4} = \frac{π}{2} + 2 πn : x = \frac{2 π}{3} + 20 + \frac{8 πn}{3}

\frac{- 60 + 3 x}{4} = \frac{π}{2} + 2 πn

Multiplicar ambos lados por

4

\frac{- 60 + 3 x}{4} = \frac{π}{2} + 2 πn

Multiplicar ambos lados por

4

\frac{4 ( - 60 + 3 x )}{4} = 4 \cdot \frac{π}{2} + 4 \cdot 2 πn

Simplificar

\frac{4 ( - 60 + 3 x )}{4} = 4 \cdot \frac{π}{2} + 4 \cdot 2 πn

Simplificar

\frac{4 ( - 60 + 3 x )}{4} : - 60 + 3 x

\frac{4 ( - 60 + 3 x )}{4}

Dividir:

\frac{4}{4} = 1

= - 60 + 3 x

Simplificar

4 \cdot \frac{π}{2} + 4 \cdot 2 πn : 2 π + 8 πn

4 \cdot \frac{π}{2} + 4 \cdot 2 πn

4 \cdot \frac{π}{2} = 2 π

4 \cdot \frac{π}{2}

Multiplicar fracciones:

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

= \frac{π 4}{2}

Dividir:

\frac{4}{2} = 2

= 2 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiplicar los numeros:

4 \cdot 2 = 8

= 8 πn

= 2 π + 8 πn

- 60 + 3 x = 2 π + 8 πn

- 60 + 3 x = 2 π + 8 πn

- 60 + 3 x = 2 π + 8 πn

Mover

60

al lado derecho

- 60 + 3 x = 2 π + 8 πn

Sumar

60

a ambos lados

- 60 + 3 x + 60 = 2 π + 8 πn + 60

Simplificar

3 x = 2 π + 8 πn + 60

3 x = 2 π + 8 πn + 60

Dividir ambos lados entre

3

3 x = 2 π + 8 πn + 60

Dividir ambos lados entre

3

\frac{3 x}{3} = \frac{2 π}{3} + \frac{8 πn}{3} + \frac{60}{3}

Simplificar

\frac{3 x}{3} = \frac{2 π}{3} + \frac{8 πn}{3} + \frac{60}{3}

Simplificar

\frac{3 x}{3} : x

\frac{3 x}{3}

Dividir:

\frac{3}{3} = 1

= x

Simplificar

\frac{2 π}{3} + \frac{8 πn}{3} + \frac{60}{3} : \frac{2 π}{3} + 20 + \frac{8 πn}{3}

\frac{2 π}{3} + \frac{8 πn}{3} + \frac{60}{3}

Agrupar términos semejantes

= \frac{2 π}{3} + \frac{60}{3} + \frac{8 πn}{3}

Dividir:

\frac{60}{3} = 20

= \frac{2 π}{3} + 20 + \frac{8 πn}{3}

x = \frac{2 π}{3} + 20 + \frac{8 πn}{3}

x = \frac{2 π}{3} + 20 + \frac{8 πn}{3}

x = \frac{2 π}{3} + 20 + \frac{8 πn}{3}

Resolver

\frac{- 60 + 3 x}{4} = \frac{3 π}{2} + 2 πn : x = 2 π + 20 + \frac{8 πn}{3}

\frac{- 60 + 3 x}{4} = \frac{3 π}{2} + 2 πn

Multiplicar ambos lados por

4

\frac{- 60 + 3 x}{4} = \frac{3 π}{2} + 2 πn

Multiplicar ambos lados por

4

\frac{4 ( - 60 + 3 x )}{4} = 4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn

Simplificar

\frac{4 ( - 60 + 3 x )}{4} = 4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn

Simplificar

\frac{4 ( - 60 + 3 x )}{4} : - 60 + 3 x

\frac{4 ( - 60 + 3 x )}{4}

Dividir:

\frac{4}{4} = 1

= - 60 + 3 x

Simplificar

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn : 6 π + 8 πn

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn

4 \cdot \frac{3 π}{2} = 6 π

4 \cdot \frac{3 π}{2}

Multiplicar fracciones:

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

= \frac{3 π 4}{2}

Multiplicar los numeros:

3 \cdot 4 = 12

= \frac{12 π}{2}

Dividir:

\frac{12}{2} = 6

= 6 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiplicar los numeros:

4 \cdot 2 = 8

= 8 πn

= 6 π + 8 πn

- 60 + 3 x = 6 π + 8 πn

- 60 + 3 x = 6 π + 8 πn

- 60 + 3 x = 6 π + 8 πn

Mover

60

al lado derecho

- 60 + 3 x = 6 π + 8 πn

Sumar

60

a ambos lados

- 60 + 3 x + 60 = 6 π + 8 πn + 60

Simplificar

3 x = 6 π + 8 πn + 60

3 x = 6 π + 8 πn + 60

Dividir ambos lados entre

3

3 x = 6 π + 8 πn + 60

Dividir ambos lados entre

3

\frac{3 x}{3} = \frac{6 π}{3} + \frac{8 πn}{3} + \frac{60}{3}

Simplificar

\frac{3 x}{3} = \frac{6 π}{3} + \frac{8 πn}{3} + \frac{60}{3}

Simplificar

\frac{3 x}{3} : x

\frac{3 x}{3}

Dividir:

\frac{3}{3} = 1

= x

Simplificar

\frac{6 π}{3} + \frac{8 πn}{3} + \frac{60}{3} : 2 π + 20 + \frac{8 πn}{3}

\frac{6 π}{3} + \frac{8 πn}{3} + \frac{60}{3}

Agrupar términos semejantes

= \frac{6 π}{3} + \frac{60}{3} + \frac{8 πn}{3}

Dividir:

\frac{6}{3} = 2

= 2 π + \frac{60}{3} + \frac{8 πn}{3}

Dividir:

\frac{60}{3} = 20

= 2 π + 20 + \frac{8 πn}{3}

x = 2 π + 20 + \frac{8 πn}{3}

x = 2 π + 20 + \frac{8 πn}{3}

x = 2 π + 20 + \frac{8 πn}{3}

x = \frac{2 π}{3} + 20 + \frac{8 πn}{3}, x = 2 π + 20 + \frac{8 πn}{3}

cos (\frac{- 60 + 5 x}{4}) = 0 : x = \frac{2 π}{5} + 12 + \frac{8 πn}{5}, x = \frac{6 π}{5} + 12 + \frac{8 πn}{5}

cos (\frac{- 60 + 5 x}{4}) = 0

Soluciones generales para

cos (\frac{- 60 + 5 x}{4}) = 0

cos (x)

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

\frac{- 60 + 5 x}{4} = \frac{π}{2} + 2 πn, \frac{- 60 + 5 x}{4} = \frac{3 π}{2} + 2 πn

\frac{- 60 + 5 x}{4} = \frac{π}{2} + 2 πn, \frac{- 60 + 5 x}{4} = \frac{3 π}{2} + 2 πn

Resolver

\frac{- 60 + 5 x}{4} = \frac{π}{2} + 2 πn : x = \frac{2 π}{5} + 12 + \frac{8 πn}{5}

\frac{- 60 + 5 x}{4} = \frac{π}{2} + 2 πn

Multiplicar ambos lados por

4

\frac{- 60 + 5 x}{4} = \frac{π}{2} + 2 πn

Multiplicar ambos lados por

4

\frac{4 ( - 60 + 5 x )}{4} = 4 \cdot \frac{π}{2} + 4 \cdot 2 πn

Simplificar

\frac{4 ( - 60 + 5 x )}{4} = 4 \cdot \frac{π}{2} + 4 \cdot 2 πn

Simplificar

\frac{4 ( - 60 + 5 x )}{4} : - 60 + 5 x

\frac{4 ( - 60 + 5 x )}{4}

Dividir:

\frac{4}{4} = 1

= - 60 + 5 x

Simplificar

4 \cdot \frac{π}{2} + 4 \cdot 2 πn : 2 π + 8 πn

4 \cdot \frac{π}{2} + 4 \cdot 2 πn

4 \cdot \frac{π}{2} = 2 π

4 \cdot \frac{π}{2}

Multiplicar fracciones:

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

= \frac{π 4}{2}

Dividir:

\frac{4}{2} = 2

= 2 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiplicar los numeros:

4 \cdot 2 = 8

= 8 πn

= 2 π + 8 πn

- 60 + 5 x = 2 π + 8 πn

- 60 + 5 x = 2 π + 8 πn

- 60 + 5 x = 2 π + 8 πn

Mover

60

al lado derecho

- 60 + 5 x = 2 π + 8 πn

Sumar

60

a ambos lados

- 60 + 5 x + 60 = 2 π + 8 πn + 60

Simplificar

5 x = 2 π + 8 πn + 60

5 x = 2 π + 8 πn + 60

Dividir ambos lados entre

5

5 x = 2 π + 8 πn + 60

Dividir ambos lados entre

5

\frac{5 x}{5} = \frac{2 π}{5} + \frac{8 πn}{5} + \frac{60}{5}

Simplificar

\frac{5 x}{5} = \frac{2 π}{5} + \frac{8 πn}{5} + \frac{60}{5}

Simplificar

\frac{5 x}{5} : x

\frac{5 x}{5}

Dividir:

\frac{5}{5} = 1

= x

Simplificar

\frac{2 π}{5} + \frac{8 πn}{5} + \frac{60}{5} : \frac{2 π}{5} + 12 + \frac{8 πn}{5}

\frac{2 π}{5} + \frac{8 πn}{5} + \frac{60}{5}

Agrupar términos semejantes

= \frac{2 π}{5} + \frac{60}{5} + \frac{8 πn}{5}

Dividir:

\frac{60}{5} = 12

= \frac{2 π}{5} + 12 + \frac{8 πn}{5}

x = \frac{2 π}{5} + 12 + \frac{8 πn}{5}

x = \frac{2 π}{5} + 12 + \frac{8 πn}{5}

x = \frac{2 π}{5} + 12 + \frac{8 πn}{5}

Resolver

\frac{- 60 + 5 x}{4} = \frac{3 π}{2} + 2 πn : x = \frac{6 π}{5} + 12 + \frac{8 πn}{5}

\frac{- 60 + 5 x}{4} = \frac{3 π}{2} + 2 πn

Multiplicar ambos lados por

4

\frac{- 60 + 5 x}{4} = \frac{3 π}{2} + 2 πn

Multiplicar ambos lados por

4

\frac{4 ( - 60 + 5 x )}{4} = 4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn

Simplificar

\frac{4 ( - 60 + 5 x )}{4} = 4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn

Simplificar

\frac{4 ( - 60 + 5 x )}{4} : - 60 + 5 x

\frac{4 ( - 60 + 5 x )}{4}

Dividir:

\frac{4}{4} = 1

= - 60 + 5 x

Simplificar

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn : 6 π + 8 πn

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn

4 \cdot \frac{3 π}{2} = 6 π

4 \cdot \frac{3 π}{2}

Multiplicar fracciones:

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

= \frac{3 π 4}{2}

Multiplicar los numeros:

3 \cdot 4 = 12

= \frac{12 π}{2}

Dividir:

\frac{12}{2} = 6

= 6 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiplicar los numeros:

4 \cdot 2 = 8

= 8 πn

= 6 π + 8 πn

- 60 + 5 x = 6 π + 8 πn

- 60 + 5 x = 6 π + 8 πn

- 60 + 5 x = 6 π + 8 πn

Mover

60

al lado derecho

- 60 + 5 x = 6 π + 8 πn

Sumar

60

a ambos lados

- 60 + 5 x + 60 = 6 π + 8 πn + 60

Simplificar

5 x = 6 π + 8 πn + 60

5 x = 6 π + 8 πn + 60

Dividir ambos lados entre

5

5 x = 6 π + 8 πn + 60

Dividir ambos lados entre

5

\frac{5 x}{5} = \frac{6 π}{5} + \frac{8 πn}{5} + \frac{60}{5}

Simplificar

\frac{5 x}{5} = \frac{6 π}{5} + \frac{8 πn}{5} + \frac{60}{5}

Simplificar

\frac{5 x}{5} : x

\frac{5 x}{5}

Dividir:

\frac{5}{5} = 1

= x

Simplificar

\frac{6 π}{5} + \frac{8 πn}{5} + \frac{60}{5} : \frac{6 π}{5} + 12 + \frac{8 πn}{5}

\frac{6 π}{5} + \frac{8 πn}{5} + \frac{60}{5}

Agrupar términos semejantes

= \frac{6 π}{5} + \frac{60}{5} + \frac{8 πn}{5}

Dividir:

\frac{60}{5} = 12

= \frac{6 π}{5} + 12 + \frac{8 πn}{5}

x = \frac{6 π}{5} + 12 + \frac{8 πn}{5}

x = \frac{6 π}{5} + 12 + \frac{8 πn}{5}

x = \frac{6 π}{5} + 12 + \frac{8 πn}{5}

x = \frac{2 π}{5} + 12 + \frac{8 πn}{5}, x = \frac{6 π}{5} + 12 + \frac{8 πn}{5}

Combinar toda las soluciones

x = \frac{2 π}{3} + 20 + \frac{8 πn}{3}, x = 2 π + 20 + \frac{8 πn}{3}, x = \frac{2 π}{5} + 12 + \frac{8 πn}{5}, x = \frac{6 π}{5} + 12 + \frac{8 πn}{5}

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