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solvefor θ,cos(θ)+cos(90-θ)=0

Solución

resolver para

θ, cos (θ) + cos (9 0^{\circ} - θ) = 0

Solución

θ = 13 5^{\circ} + 36 0^{\circ} n, θ = 31 5^{\circ} + 36 0^{\circ} n

radianes

θ = \frac{3 π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

Pasos de solución

cos (θ) + cos (9 0^{\circ} - θ) = 0

Re-escribir usando identidades trigonométricas

cos (θ) + cos (9 0^{\circ} - θ)

Utilizar la identidad suma-producto:

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

= 2 cos (\frac{θ + 9 0 ^{\circ} - θ}{2}) cos (\frac{θ - ( 9 0 ^{\circ} - θ )}{2})

Simplificar

2 cos (\frac{θ + 9 0 ^{\circ} - θ}{2}) cos (\frac{θ - ( 9 0 ^{\circ} - θ )}{2}) : 2 cos (\frac{4 θ - 18 0 ^{\circ}}{4})

2 cos (\frac{θ + 9 0 ^{\circ} - θ}{2}) cos (\frac{θ - ( 9 0 ^{\circ} - θ )}{2})

\frac{θ + 9 0 ^{\circ} - θ}{2} = 4 5^{\circ}

\frac{θ + 9 0 ^{\circ} - θ}{2}

θ + 9 0^{\circ} - θ = 9 0^{\circ}

θ + 9 0^{\circ} - θ

Agrupar términos semejantes

= θ - θ + 9 0^{\circ}

Sumar elementos similares:

θ - θ = 0

= 9 0^{\circ}

= \frac{9 0 ^{\circ}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{18 0 ^{\circ}}{2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= 4 5^{\circ}

= 2 cos (4 5^{\circ}) cos (\frac{θ - ( - θ + 9 0 ^{\circ} )}{2})

\frac{θ - ( 9 0 ^{\circ} - θ )}{2} = \frac{4 θ - 18 0 ^{\circ}}{4}

\frac{θ - ( 9 0 ^{\circ} - θ )}{2}

Expandir

θ - (9 0^{\circ} - θ) : 2 θ - 9 0^{\circ}

θ - (9 0^{\circ} - θ)

- (9 0^{\circ} - θ) : - 9 0^{\circ} + θ

- (9 0^{\circ} - θ)

Poner los parentesis

= - (9 0^{\circ}) - (- θ)

Aplicar las reglas de los signos

- (- a) = a, - (a) = - a

= - 9 0^{\circ} + θ

= θ - 9 0^{\circ} + θ

Simplificar

θ - 9 0^{\circ} + θ : 2 θ - 9 0^{\circ}

θ - 9 0^{\circ} + θ

Agrupar términos semejantes

= θ + θ - 9 0^{\circ}

Sumar elementos similares:

θ + θ = 2 θ

= 2 θ - 9 0^{\circ}

= 2 θ - 9 0^{\circ}

= \frac{2 θ - 9 0 ^{\circ}}{2}

Simplificar

2 θ - 9 0^{\circ}

en una fracción:

\frac{4 θ - 18 0 ^{\circ}}{2}

2 θ - 9 0^{\circ}

Convertir a fracción:

2 θ = \frac{2 θ 2}{2}

= \frac{2 θ \cdot 2}{2} - 9 0^{\circ}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{2 θ \cdot 2 - 18 0 ^{\circ}}{2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{4 θ - 18 0 ^{\circ}}{2}

= \frac{\frac{4 θ - 18 0 ^{\circ}}{2}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{4 θ - 18 0 ^{\circ}}{2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{4 θ - 18 0 ^{\circ}}{4}

= 2 cos (4 5^{\circ}) cos (\frac{4 θ - 18 0 ^{\circ}}{4})

Simplificar

cos (4 5^{\circ}) : \frac{2}{2}

cos (4 5^{\circ})

Utilizar la siguiente identidad trivial:

cos (4 5^{\circ}) = \frac{2}{2}

cos (x)

tabla de valores periódicos con

36 0^{\circ} n

intervalos:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

= 2 \cdot \frac{2}{2} cos (\frac{4 θ - 18 0 ^{\circ}}{4})

Multiplicar fracciones:

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

= \frac{2 \cdot 2 cos ( \frac{4 θ - 18 0 ^{\circ}}{4} )}{2}

Eliminar los terminos comunes:

2

= 2 cos (\frac{4 θ - 18 0 ^{\circ}}{4})

= 2 cos (\frac{4 θ - 18 0 ^{\circ}}{4})

2 cos (\frac{4 θ - 18 0 ^{\circ}}{4}) = 0

Dividir ambos lados entre

2

2 cos (\frac{4 θ - 18 0 ^{\circ}}{4}) = 0

Dividir ambos lados entre

2

\frac{2 cos ( \frac{4 θ - 18 0 ^{\circ}}{4} )}{2} = \frac{0}{2}

Simplificar

cos (\frac{4 θ - 18 0 ^{\circ}}{4}) = 0

cos (\frac{4 θ - 18 0 ^{\circ}}{4}) = 0

Soluciones generales para

cos (\frac{4 θ - 18 0 ^{\circ}}{4}) = 0

cos (x)

tabla de valores periódicos con

36 0^{\circ} n

intervalos:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

\frac{4 θ - 18 0 ^{\circ}}{4} = 9 0^{\circ} + 36 0^{\circ} n, \frac{4 θ - 18 0 ^{\circ}}{4} = 27 0^{\circ} + 36 0^{\circ} n

\frac{4 θ - 18 0 ^{\circ}}{4} = 9 0^{\circ} + 36 0^{\circ} n, \frac{4 θ - 18 0 ^{\circ}}{4} = 27 0^{\circ} + 36 0^{\circ} n

Resolver

\frac{4 θ - 18 0 ^{\circ}}{4} = 9 0^{\circ} + 36 0^{\circ} n : θ = 13 5^{\circ} + 36 0^{\circ} n

\frac{4 θ - 18 0 ^{\circ}}{4} = 9 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

4

\frac{4 θ - 18 0 ^{\circ}}{4} = 9 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

4

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4} = 4 \cdot 9 0^{\circ} + 4 \cdot 36 0^{\circ} n

Simplificar

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4} = 4 \cdot 9 0^{\circ} + 4 \cdot 36 0^{\circ} n

Simplificar

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4} : 4 θ - 18 0^{\circ}

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4}

Dividir:

\frac{4}{4} = 1

= 4 θ - 18 0^{\circ}

Simplificar

4 \cdot 9 0^{\circ} + 4 \cdot 36 0^{\circ} n : 36 0^{\circ} + 144 0^{\circ} n

4 \cdot 9 0^{\circ} + 4 \cdot 36 0^{\circ} n

4 \cdot 9 0^{\circ} = 36 0^{\circ}

4 \cdot 9 0^{\circ}

Multiplicar fracciones:

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

= 36 0^{\circ}

Dividir:

\frac{4}{2} = 2

= 36 0^{\circ}

4 \cdot 36 0^{\circ} n = 144 0^{\circ} n

4 \cdot 36 0^{\circ} n

Multiplicar los numeros:

4 \cdot 2 = 8

= 144 0^{\circ} n

= 36 0^{\circ} + 144 0^{\circ} n

4 θ - 18 0^{\circ} = 36 0^{\circ} + 144 0^{\circ} n

4 θ - 18 0^{\circ} = 36 0^{\circ} + 144 0^{\circ} n

4 θ - 18 0^{\circ} = 36 0^{\circ} + 144 0^{\circ} n

Mover

18 0^{\circ}

al lado derecho

4 θ - 18 0^{\circ} = 36 0^{\circ} + 144 0^{\circ} n

Sumar

18 0^{\circ}

a ambos lados

4 θ - 18 0^{\circ} + 18 0^{\circ} = 36 0^{\circ} + 144 0^{\circ} n + 18 0^{\circ}

Simplificar

4 θ = 54 0^{\circ} + 144 0^{\circ} n

4 θ = 54 0^{\circ} + 144 0^{\circ} n

Dividir ambos lados entre

4

4 θ = 54 0^{\circ} + 144 0^{\circ} n

Dividir ambos lados entre

4

\frac{4 θ}{4} = 13 5^{\circ} + \frac{144 0 ^{\circ} n}{4}

Simplificar

θ = 13 5^{\circ} + 36 0^{\circ} n

θ = 13 5^{\circ} + 36 0^{\circ} n

Resolver

\frac{4 θ - 18 0 ^{\circ}}{4} = 27 0^{\circ} + 36 0^{\circ} n : θ = 31 5^{\circ} + 36 0^{\circ} n

\frac{4 θ - 18 0 ^{\circ}}{4} = 27 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

4

\frac{4 θ - 18 0 ^{\circ}}{4} = 27 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

4

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4} = 4 \cdot 27 0^{\circ} + 4 \cdot 36 0^{\circ} n

Simplificar

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4} = 4 \cdot 27 0^{\circ} + 4 \cdot 36 0^{\circ} n

Simplificar

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4} : 4 θ - 18 0^{\circ}

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4}

Dividir:

\frac{4}{4} = 1

= 4 θ - 18 0^{\circ}

Simplificar

4 \cdot 27 0^{\circ} + 4 \cdot 36 0^{\circ} n : 108 0^{\circ} + 144 0^{\circ} n

4 \cdot 27 0^{\circ} + 4 \cdot 36 0^{\circ} n

4 \cdot 27 0^{\circ} = 108 0^{\circ}

4 \cdot 27 0^{\circ}

Multiplicar fracciones:

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

= 108 0^{\circ}

Multiplicar los numeros:

3 \cdot 4 = 12

= 108 0^{\circ}

Dividir:

\frac{12}{2} = 6

= 108 0^{\circ}

4 \cdot 36 0^{\circ} n = 144 0^{\circ} n

4 \cdot 36 0^{\circ} n

Multiplicar los numeros:

4 \cdot 2 = 8

= 144 0^{\circ} n

= 108 0^{\circ} + 144 0^{\circ} n

4 θ - 18 0^{\circ} = 108 0^{\circ} + 144 0^{\circ} n

4 θ - 18 0^{\circ} = 108 0^{\circ} + 144 0^{\circ} n

4 θ - 18 0^{\circ} = 108 0^{\circ} + 144 0^{\circ} n

Mover

18 0^{\circ}

al lado derecho

4 θ - 18 0^{\circ} = 108 0^{\circ} + 144 0^{\circ} n

Sumar

18 0^{\circ}

a ambos lados

4 θ - 18 0^{\circ} + 18 0^{\circ} = 108 0^{\circ} + 144 0^{\circ} n + 18 0^{\circ}

Simplificar

4 θ = 126 0^{\circ} + 144 0^{\circ} n

4 θ = 126 0^{\circ} + 144 0^{\circ} n

Dividir ambos lados entre

4

4 θ = 126 0^{\circ} + 144 0^{\circ} n

Dividir ambos lados entre

4

\frac{4 θ}{4} = 31 5^{\circ} + \frac{144 0 ^{\circ} n}{4}

Simplificar

θ = 31 5^{\circ} + 36 0^{\circ} n

θ = 31 5^{\circ} + 36 0^{\circ} n

θ = 13 5^{\circ} + 36 0^{\circ} n, θ = 31 5^{\circ} + 36 0^{\circ} n

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