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Popular Trigonometría >

cos(2x-180)-sin(x-90)=0

Solución

cos (2 x - 18 0^{\circ}) - sin (x - 9 0^{\circ}) = 0

Solución

x = \frac{72 0 ^{\circ} n}{3}, x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

Radianes

x = 0 + \frac{4 π}{3} n, x = \frac{2 π}{3} + \frac{4 π}{3} n

Pasos de solución

cos (2 x - 18 0^{\circ}) - sin (x - 9 0^{\circ}) = 0

Re-escribir usando identidades trigonométricas

cos (2 x - 18 0^{\circ}) - sin (x - 9 0^{\circ}) = 0

Re-escribir usando identidades trigonométricas

sin (x - 9 0^{\circ})

Utilizar la identidad de diferencia de ángulos:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= sin (x) cos (9 0^{\circ}) - cos (x) sin (9 0^{\circ})

Simplificar

sin (x) cos (9 0^{\circ}) - cos (x) sin (9 0^{\circ}) : - cos (x)

sin (x) cos (9 0^{\circ}) - cos (x) sin (9 0^{\circ})

sin (x) cos (9 0^{\circ}) = 0

sin (x) cos (9 0^{\circ})

Simplificar

cos (9 0^{\circ}) : 0

cos (9 0^{\circ})

Utilizar la siguiente identidad trivial:

cos (9 0^{\circ}) = 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}

= 0

= 0 \cdot sin (x)

Aplicar la regla

0 \cdot a = 0

= 0

cos (x) sin (9 0^{\circ}) = cos (x)

cos (x) sin (9 0^{\circ})

Simplificar

sin (9 0^{\circ}) : 1

sin (9 0^{\circ})

Utilizar la siguiente identidad trivial:

sin (9 0^{\circ}) = 1

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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 1

= 1 \cdot cos (x)

Multiplicar:

cos (x) \cdot 1 = cos (x)

= cos (x)

= 0 - cos (x)

0 - cos (x) = - cos (x)

= - cos (x)

= - cos (x)

Utilizar la identidad de diferencia de ángulos:

cos (s - t) = cos (s) cos (t) + sin (s) sin (t)

= cos (2 x) cos (18 0^{\circ}) + sin (2 x) sin (18 0^{\circ})

Simplificar

cos (2 x) cos (18 0^{\circ}) + sin (2 x) sin (18 0^{\circ}) : - cos (2 x)

cos (2 x) cos (18 0^{\circ}) + sin (2 x) sin (18 0^{\circ})

cos (2 x) cos (18 0^{\circ}) = - cos (2 x)

cos (2 x) cos (18 0^{\circ})

Simplificar

cos (18 0^{\circ}) : - 1

cos (18 0^{\circ})

Utilizar la siguiente identidad trivial:

cos (18 0^{\circ}) = (- 1)

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}

= - 1

= (- 1) cos (2 x)

Simplificar

= - cos (2 x)

= - cos (2 x) + sin (18 0^{\circ}) sin (2 x)

sin (2 x) sin (18 0^{\circ}) = 0

sin (2 x) sin (18 0^{\circ})

Simplificar

sin (18 0^{\circ}) : 0

sin (18 0^{\circ})

Utilizar la siguiente identidad trivial:

sin (18 0^{\circ}) = 0

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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

= 0 \cdot sin (2 x)

Aplicar la regla

0 \cdot a = 0

= 0

= - cos (2 x) + 0

- cos (2 x) + 0 = - cos (2 x)

= - cos (2 x)

= - cos (2 x)

- cos (2 x) - (- cos (x)) = 0

Simplificar

- cos (2 x) - (- cos (x)) : - cos (2 x) + cos (x)

- cos (2 x) - (- cos (x))

Aplicar la regla

- (- a) = a

= - cos (2 x) + cos (x)

- cos (2 x) + cos (x) = 0

Utilizar la identidad suma-producto:

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

= - 2 sin (\frac{x + 2 x}{2}) sin (\frac{x - 2 x}{2})

Simplificar

- 2 sin (\frac{x + 2 x}{2}) sin (\frac{x - 2 x}{2}) : 2 sin (\frac{x}{2}) sin (\frac{3 x}{2})

- 2 sin (\frac{x + 2 x}{2}) sin (\frac{x - 2 x}{2})

Sumar elementos similares:

x + 2 x = 3 x

= - 2 sin (\frac{3 x}{2}) sin (\frac{x - 2 x}{2})

\frac{x - 2 x}{2} = - \frac{x}{2}

\frac{x - 2 x}{2}

Sumar elementos similares:

x - 2 x = - x

= \frac{- x}{2}

Aplicar las propiedades de las fracciones:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{x}{2}

= - 2 sin (\frac{3 x}{2}) sin (- \frac{x}{2})

Utilizar la razón trigonométrica de ángulo negativo:

sin (- x) = - sin (x)

= - 2 (- sin (\frac{x}{2})) sin (\frac{3 x}{2})

Aplicar la regla

- (- a) = a

= 2 sin (\frac{x}{2}) sin (\frac{3 x}{2})

= 2 sin (\frac{x}{2}) sin (\frac{3 x}{2})

2 sin (\frac{3 x}{2}) sin (\frac{x}{2}) = 0

Resolver cada parte por separado

sin (\frac{3 x}{2}) = 0 or sin (\frac{x}{2}) = 0

sin (\frac{3 x}{2}) = 0 : x = \frac{72 0 ^{\circ} n}{3}, x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

sin (\frac{3 x}{2}) = 0

Soluciones generales para

sin (\frac{3 x}{2}) = 0

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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{3 x}{2} = 0 + 36 0^{\circ} n, \frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n

\frac{3 x}{2} = 0 + 36 0^{\circ} n, \frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n

Resolver

\frac{3 x}{2} = 0 + 36 0^{\circ} n : x = \frac{72 0 ^{\circ} n}{3}

\frac{3 x}{2} = 0 + 36 0^{\circ} n

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

\frac{3 x}{2} = 36 0^{\circ} n

Multiplicar ambos lados por

2

\frac{3 x}{2} = 36 0^{\circ} n

Multiplicar ambos lados por

2

\frac{2 \cdot 3 x}{2} = 2 \cdot 36 0^{\circ} n

Simplificar

3 x = 72 0^{\circ} n

3 x = 72 0^{\circ} n

Dividir ambos lados entre

3

3 x = 72 0^{\circ} n

Dividir ambos lados entre

3

\frac{3 x}{3} = \frac{72 0 ^{\circ} n}{3}

Simplificar

x = \frac{72 0 ^{\circ} n}{3}

x = \frac{72 0 ^{\circ} n}{3}

Resolver

\frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n : x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

\frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

2

\frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

2

\frac{2 \cdot 3 x}{2} = 36 0^{\circ} + 2 \cdot 36 0^{\circ} n

Simplificar

3 x = 36 0^{\circ} + 72 0^{\circ} n

3 x = 36 0^{\circ} + 72 0^{\circ} n

Dividir ambos lados entre

3

3 x = 36 0^{\circ} + 72 0^{\circ} n

Dividir ambos lados entre

3

\frac{3 x}{3} = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

Simplificar

x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

x = \frac{72 0 ^{\circ} n}{3}, x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

sin (\frac{x}{2}) = 0 : x = 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n

sin (\frac{x}{2}) = 0

Soluciones generales para

sin (\frac{x}{2}) = 0

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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{x}{2} = 0 + 36 0^{\circ} n, \frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n

\frac{x}{2} = 0 + 36 0^{\circ} n, \frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n

Resolver

\frac{x}{2} = 0 + 36 0^{\circ} n : x = 72 0^{\circ} n

\frac{x}{2} = 0 + 36 0^{\circ} n

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

\frac{x}{2} = 36 0^{\circ} n

Multiplicar ambos lados por

2

\frac{x}{2} = 36 0^{\circ} n

Multiplicar ambos lados por

2

\frac{2 x}{2} = 2 \cdot 36 0^{\circ} n

Simplificar

x = 72 0^{\circ} n

x = 72 0^{\circ} n

Resolver

\frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} + 72 0^{\circ} n

\frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

2

\frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

2

\frac{2 x}{2} = 36 0^{\circ} + 2 \cdot 36 0^{\circ} n

Simplificar

x = 36 0^{\circ} + 72 0^{\circ} n

x = 36 0^{\circ} + 72 0^{\circ} n

x = 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n

Combinar toda las soluciones

x = \frac{72 0 ^{\circ} n}{3}, x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}, x = 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n

Mezclar intervalos sobrepuestos

x = \frac{72 0 ^{\circ} n}{3}, x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

Gráfica

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