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sin^2(x+30)<sin^2(x)

Solución

sin^{2} (x + 3 0^{\circ}) < sin^{2} (x)

Solución

7 5^{\circ} + πn < x < 16 5^{\circ} + πn

Notación de intervalos

(7 5^{\circ} + πn, 16 5^{\circ} + πn)

Notación decimal

1.30899 \dots + πn < x < 2.87979 \dots + πn

Pasos de solución

sin^{2} (x + 3 0^{\circ}) < sin^{2} (x)

Mover

sin^{2} (x)

al lado izquierdo

sin^{2} (x + 3 0^{\circ}) < sin^{2} (x)

Restar

sin^{2} (x)

de ambos lados

sin^{2} (x + 3 0^{\circ}) - sin^{2} (x) < sin^{2} (x) - sin^{2} (x)

sin^{2} (x + 3 0^{\circ}) - sin^{2} (x) < 0

sin^{2} (x + 3 0^{\circ}) - sin^{2} (x) < 0

Usar la siguiente identidad:

cos^{2} (x) + sin^{2} (x) = 1

Por lo tanto

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

1 - cos^{2} (x + 3 0^{\circ}) - (1 - cos^{2} (x)) < 0

Simplificar

cos^{2} (x) - cos^{2} (x + 3 0^{\circ}) < 0

Periodicidad de

cos^{2} (x) - cos^{2} (x + 3 0^{\circ}) : π

La periodicidad combinada de la suma de funciones periódicas es el mínimo común múltiplo de los períodos

cos^{2} (x), cos^{2} (x + 3 0^{\circ})

Periodicidad de

cos^{2} (x) : π

Periodicidad de

cos^{n} (x) = \frac{Periodicidad de cos ( x )}{2},

si n es par

Periodicidad de

cos (x) : 2 π

La periodicidad de

: cos (x)

2 π

= 2 π

\frac{2 π}{2}

Simplificar

π

Periodicidad de

cos^{2} (x + 3 0^{\circ}) : π

Periodicidad de

cos^{n} (x) = \frac{Periodicidad de cos ( x )}{2},

si n es par

Periodicidad de

cos (x + 3 0^{\circ}) : 2 π

La periodicidad de

: cos (x)

2 π

= 2 π

\frac{2 π}{2}

Simplificar

π

Combinar períodos:

π, π

= π

Factorizar

cos^{2} (x) - cos^{2} (x + 3 0^{\circ}) : (cos (x) + cos (x + 3 0^{\circ})) (cos (x) - cos (x + 3 0^{\circ}))

cos^{2} (x) - cos^{2} (x + 3 0^{\circ})

Aplicar la siguiente regla para binomios al cuadrado:

x^{2} - y^{2} = (x + y) (x - y)

cos^{2} (x) - cos^{2} (x + 3 0^{\circ}) = (cos (x) + cos (x + 3 0^{\circ})) (cos (x) - cos (x + 3 0^{\circ}))

= (cos (x) + cos (x + 3 0^{\circ})) (cos (x) - cos (x + 3 0^{\circ}))

(cos (x) + cos (x + 3 0^{\circ})) (cos (x) - cos (x + 3 0^{\circ})) < 0

Para encontrar los ceros, transformar la desigualdad a 0

(cos (x) + cos (x + 3 0^{\circ})) (cos (x) - cos (x + 3 0^{\circ})) = 0

Resolver

(cos (x) + cos (x + 3 0^{\circ})) (cos (x) - cos (x + 3 0^{\circ})) = 0

para

0 \leq x < π

(cos (x) + cos (x + 3 0^{\circ})) (cos (x) - cos (x + 3 0^{\circ})) = 0

Resolver cada parte por separado

cos (x) + cos (x + 3 0^{\circ}) = 0 : x = 7 5^{\circ}

cos (x) + cos (x + 3 0^{\circ}) = 0, 0 \leq x < π

Re-escribir usando identidades trigonométricas

cos (x) + cos (x + 3 0^{\circ})

Utilizar la identidad suma-producto:

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

= 2 cos (\frac{x + x + 3 0 ^{\circ}}{2}) cos (\frac{x - ( x + 3 0 ^{\circ} )}{2})

Simplificar

2 cos (\frac{x + x + 3 0 ^{\circ}}{2}) cos (\frac{x - ( x + 3 0 ^{\circ} )}{2}) : \frac{6 + 2}{2} cos (\frac{12 x + 18 0 ^{\circ}}{12})

2 cos (\frac{x + x + 3 0 ^{\circ}}{2}) cos (\frac{x - ( x + 3 0 ^{\circ} )}{2})

\frac{x + x + 3 0 ^{\circ}}{2} = \frac{12 x + 18 0 ^{\circ}}{12}

\frac{x + x + 3 0 ^{\circ}}{2}

Sumar elementos similares:

x + x = 2 x

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

Simplificar

2 x + 3 0^{\circ}

en una fracción:

\frac{12 x + 18 0 ^{\circ}}{6}

2 x + 3 0^{\circ}

Convertir a fracción:

2 x = \frac{2 x 6}{6}

= \frac{2 x \cdot 6}{6} + 3 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 x \cdot 6 + 18 0 ^{\circ}}{6}

Multiplicar los numeros:

2 \cdot 6 = 12

= \frac{12 x + 18 0 ^{\circ}}{6}

= \frac{\frac{12 x + 18 0 ^{\circ}}{6}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{12 x + 18 0 ^{\circ}}{6 \cdot 2}

Multiplicar los numeros:

6 \cdot 2 = 12

= \frac{12 x + 18 0 ^{\circ}}{12}

= 2 cos (\frac{12 x + 18 0 ^{\circ}}{12}) cos (\frac{x - ( x + 3 0 ^{\circ} )}{2})

\frac{x - ( x + 3 0 ^{\circ} )}{2} = - 1 5^{\circ}

\frac{x - ( x + 3 0 ^{\circ} )}{2}

Expandir

x - (x + 3 0^{\circ}) : - 3 0^{\circ}

x - (x + 3 0^{\circ})

- (x + 3 0^{\circ}) : - x - 3 0^{\circ}

- (x + 3 0^{\circ})

Poner los parentesis

= - (x) - (3 0^{\circ})

Aplicar las reglas de los signos

+ (- a) = - a

= - x - 3 0^{\circ}

= x - x - 3 0^{\circ}

Sumar elementos similares:

x - x = 0

= - 3 0^{\circ}

= \frac{- 3 0 ^{\circ}}{2}

Aplicar las propiedades de las fracciones:

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

= - \frac{3 0 ^{\circ}}{2}

Aplicar las propiedades de las fracciones:

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

\frac{3 0 ^{\circ}}{2} = \frac{18 0 ^{\circ}}{6 \cdot 2}

= - \frac{18 0 ^{\circ}}{6 \cdot 2}

Multiplicar los numeros:

6 \cdot 2 = 12

= - 1 5^{\circ}

= 2 cos (- 1 5^{\circ}) cos (\frac{12 x + 18 0 ^{\circ}}{12})

cos (- 1 5^{\circ}) = \frac{6 + 2}{4}

cos (- 1 5^{\circ})

Utilizar la siguiente propiedad:

cos (- x) = cos (x)

cos (- 1 5^{\circ}) = cos (1 5^{\circ})

= cos (1 5^{\circ})

Re-escribir usando identidades trigonométricas:

cos (4 5^{\circ}) cos (3 0^{\circ}) + sin (4 5^{\circ}) sin (3 0^{\circ})

cos (1 5^{\circ})

Escribir

cos (1 5^{\circ})

como

cos (4 5^{\circ} - 3 0^{\circ})

= cos (4 5^{\circ} - 3 0^{\circ})

Utilizar la identidad de diferencia de ángulos:

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

= cos (4 5^{\circ}) cos (3 0^{\circ}) + sin (4 5^{\circ}) sin (3 0^{\circ})

= cos (4 5^{\circ}) cos (3 0^{\circ}) + sin (4 5^{\circ}) sin (3 0^{\circ})

Utilizar la siguiente identidad trivial:

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

cos (4 5^{\circ})

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}

Utilizar la siguiente identidad trivial:

cos (3 0^{\circ}) = \frac{3}{2}

cos (3 0^{\circ})

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{3}{2}

Utilizar la siguiente identidad trivial:

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

sin (4 5^{\circ})

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{2}{2}

Utilizar la siguiente identidad trivial:

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 0^{\circ})

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{1}{2}

= \frac{2}{2} \cdot \frac{3}{2} + \frac{2}{2} \cdot \frac{1}{2}

Simplificar

\frac{2}{2} \cdot \frac{3}{2} + \frac{2}{2} \cdot \frac{1}{2} : \frac{6 + 2}{4}

\frac{2}{2} \cdot \frac{3}{2} + \frac{2}{2} \cdot \frac{1}{2}

\frac{2}{2} \cdot \frac{3}{2} = \frac{6}{4}

\frac{2}{2} \cdot \frac{3}{2}

Multiplicar fracciones:

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

= \frac{2 3}{2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{2 3}{4}

Simplificar

23 : 6

23

Aplicar las leyes de los exponentes:

a b = a \cdot b

23 = 2 \cdot 3

= 2 \cdot 3

Multiplicar los numeros:

2 \cdot 3 = 6

= 6

= \frac{6}{4}

\frac{2}{2} \cdot \frac{1}{2} = \frac{2}{4}

\frac{2}{2} \cdot \frac{1}{2}

Multiplicar fracciones:

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

= \frac{2 \cdot 1}{2 \cdot 2}

Simplificar

= \frac{2}{4}

= \frac{6}{4} + \frac{2}{4}

Aplicar la regla

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

= \frac{6 + 2}{4}

= \frac{6 + 2}{4}

= 2 \cdot \frac{6 + 2}{4} cos (\frac{12 x + 18 0 ^{\circ}}{12})

Multiplicar fracciones:

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

= \frac{2 ( 6 + 2 )}{4} cos (\frac{12 x + 18 0 ^{\circ}}{12})

Eliminar los terminos comunes:

2

= \frac{6 + 2}{2} cos (\frac{12 x + 18 0 ^{\circ}}{12})

= \frac{6 + 2}{2} cos (\frac{12 x + 18 0 ^{\circ}}{12})

\frac{6 + 2}{2} cos (\frac{12 x + 18 0 ^{\circ}}{12}) = 0

Multiplicar ambos lados por

2

\frac{6 + 2}{2} cos (\frac{12 x + 18 0 ^{\circ}}{12}) = 0

Multiplicar ambos lados por

2

2 \cdot \frac{6 + 2}{2} cos (\frac{12 x + 18 0 ^{\circ}}{12}) = 0 \cdot 2

Simplificar

(6 + 2) cos (\frac{12 x + 18 0 ^{\circ}}{12}) = 0

(6 + 2) cos (\frac{12 x + 18 0 ^{\circ}}{12}) = 0

Dividir ambos lados entre

6 + 2

(6 + 2) cos (\frac{12 x + 18 0 ^{\circ}}{12}) = 0

Dividir ambos lados entre

6 + 2

\frac{( 6 + 2 ) cos ( \frac{12 x + 18 0 ^{\circ}}{12} )}{6 + 2} = \frac{0}{6 + 2}

Simplificar

cos (\frac{12 x + 18 0 ^{\circ}}{12}) = 0

cos (\frac{12 x + 18 0 ^{\circ}}{12}) = 0

Soluciones generales para

cos (\frac{12 x + 18 0 ^{\circ}}{12}) = 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{12 x + 18 0 ^{\circ}}{12} = 9 0^{\circ} + 36 0^{\circ} n, \frac{12 x + 18 0 ^{\circ}}{12} = 27 0^{\circ} + 36 0^{\circ} n

\frac{12 x + 18 0 ^{\circ}}{12} = 9 0^{\circ} + 36 0^{\circ} n, \frac{12 x + 18 0 ^{\circ}}{12} = 27 0^{\circ} + 36 0^{\circ} n

Resolver

\frac{12 x + 18 0 ^{\circ}}{12} = 9 0^{\circ} + 36 0^{\circ} n : x = 7 5^{\circ} + 36 0^{\circ} n

\frac{12 x + 18 0 ^{\circ}}{12} = 9 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

12

\frac{12 x + 18 0 ^{\circ}}{12} = 9 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

12

\frac{12 ( 12 x + 18 0 ^{\circ} )}{12} = 12 \cdot 9 0^{\circ} + 12 \cdot 36 0^{\circ} n

Simplificar

\frac{12 ( 12 x + 18 0 ^{\circ} )}{12} = 12 \cdot 9 0^{\circ} + 12 \cdot 36 0^{\circ} n

Simplificar

\frac{12 ( 12 x + 18 0 ^{\circ} )}{12} : 12 x + 18 0^{\circ}

\frac{12 ( 12 x + 18 0 ^{\circ} )}{12}

Dividir:

\frac{12}{12} = 1

= 12 x + 18 0^{\circ}

Simplificar

12 \cdot 9 0^{\circ} + 12 \cdot 36 0^{\circ} n : 108 0^{\circ} + 432 0^{\circ} n

12 \cdot 9 0^{\circ} + 12 \cdot 36 0^{\circ} n

12 \cdot 9 0^{\circ} = 108 0^{\circ}

12 \cdot 9 0^{\circ}

Multiplicar fracciones:

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

= 108 0^{\circ}

Dividir:

\frac{12}{2} = 6

= 108 0^{\circ}

12 \cdot 36 0^{\circ} n = 432 0^{\circ} n

12 \cdot 36 0^{\circ} n

Multiplicar los numeros:

12 \cdot 2 = 24

= 432 0^{\circ} n

= 108 0^{\circ} + 432 0^{\circ} n

12 x + 18 0^{\circ} = 108 0^{\circ} + 432 0^{\circ} n

12 x + 18 0^{\circ} = 108 0^{\circ} + 432 0^{\circ} n

12 x + 18 0^{\circ} = 108 0^{\circ} + 432 0^{\circ} n

Mover

18 0^{\circ}

al lado derecho

12 x + 18 0^{\circ} = 108 0^{\circ} + 432 0^{\circ} n

Restar

18 0^{\circ}

de ambos lados

12 x + 18 0^{\circ} - 18 0^{\circ} = 108 0^{\circ} + 432 0^{\circ} n - 18 0^{\circ}

Simplificar

12 x = 90 0^{\circ} + 432 0^{\circ} n

12 x = 90 0^{\circ} + 432 0^{\circ} n

Dividir ambos lados entre

12

12 x = 90 0^{\circ} + 432 0^{\circ} n

Dividir ambos lados entre

12

\frac{12 x}{12} = 7 5^{\circ} + \frac{432 0 ^{\circ} n}{12}

Simplificar

x = 7 5^{\circ} + 36 0^{\circ} n

x = 7 5^{\circ} + 36 0^{\circ} n

Resolver

\frac{12 x + 18 0 ^{\circ}}{12} = 27 0^{\circ} + 36 0^{\circ} n : x = 25 5^{\circ} + 36 0^{\circ} n

\frac{12 x + 18 0 ^{\circ}}{12} = 27 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

12

\frac{12 x + 18 0 ^{\circ}}{12} = 27 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

12

\frac{12 ( 12 x + 18 0 ^{\circ} )}{12} = 12 \cdot 27 0^{\circ} + 12 \cdot 36 0^{\circ} n

Simplificar

\frac{12 ( 12 x + 18 0 ^{\circ} )}{12} = 12 \cdot 27 0^{\circ} + 12 \cdot 36 0^{\circ} n

Simplificar

\frac{12 ( 12 x + 18 0 ^{\circ} )}{12} : 12 x + 18 0^{\circ}

\frac{12 ( 12 x + 18 0 ^{\circ} )}{12}

Dividir:

\frac{12}{12} = 1

= 12 x + 18 0^{\circ}

Simplificar

12 \cdot 27 0^{\circ} + 12 \cdot 36 0^{\circ} n : 324 0^{\circ} + 432 0^{\circ} n

12 \cdot 27 0^{\circ} + 12 \cdot 36 0^{\circ} n

12 \cdot 27 0^{\circ} = 324 0^{\circ}

12 \cdot 27 0^{\circ}

Multiplicar fracciones:

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

= 324 0^{\circ}

Multiplicar los numeros:

3 \cdot 12 = 36

= 324 0^{\circ}

Dividir:

\frac{36}{2} = 18

= 324 0^{\circ}

12 \cdot 36 0^{\circ} n = 432 0^{\circ} n

12 \cdot 36 0^{\circ} n

Multiplicar los numeros:

12 \cdot 2 = 24

= 432 0^{\circ} n

= 324 0^{\circ} + 432 0^{\circ} n

12 x + 18 0^{\circ} = 324 0^{\circ} + 432 0^{\circ} n

12 x + 18 0^{\circ} = 324 0^{\circ} + 432 0^{\circ} n

12 x + 18 0^{\circ} = 324 0^{\circ} + 432 0^{\circ} n

Mover

18 0^{\circ}

al lado derecho

12 x + 18 0^{\circ} = 324 0^{\circ} + 432 0^{\circ} n

Restar

18 0^{\circ}

de ambos lados

12 x + 18 0^{\circ} - 18 0^{\circ} = 324 0^{\circ} + 432 0^{\circ} n - 18 0^{\circ}

Simplificar

12 x = 306 0^{\circ} + 432 0^{\circ} n

12 x = 306 0^{\circ} + 432 0^{\circ} n

Dividir ambos lados entre

12

12 x = 306 0^{\circ} + 432 0^{\circ} n

Dividir ambos lados entre

12

\frac{12 x}{12} = 25 5^{\circ} + \frac{432 0 ^{\circ} n}{12}

Simplificar

x = 25 5^{\circ} + 36 0^{\circ} n

x = 25 5^{\circ} + 36 0^{\circ} n

x = 7 5^{\circ} + 36 0^{\circ} n, x = 25 5^{\circ} + 36 0^{\circ} n

Soluciones para el rango

0 \leq x < 18 0^{\circ}

x = 7 5^{\circ}

cos (x) - cos (x + 3 0^{\circ}) = 0 : x = 16 5^{\circ}

cos (x) - cos (x + 3 0^{\circ}) = 0, 0 \leq x < π

Re-escribir usando identidades trigonométricas

cos (x) - cos (x + 3 0^{\circ})

Utilizar la identidad suma-producto:

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

= - 2 sin (\frac{x + x + 3 0 ^{\circ}}{2}) sin (\frac{x - ( x + 3 0 ^{\circ} )}{2})

Simplificar

- 2 sin (\frac{x + x + 3 0 ^{\circ}}{2}) sin (\frac{x - ( x + 3 0 ^{\circ} )}{2}) : \frac{6 - 2}{2} sin (\frac{12 x + 18 0 ^{\circ}}{12})

- 2 sin (\frac{x + x + 3 0 ^{\circ}}{2}) sin (\frac{x - ( x + 3 0 ^{\circ} )}{2})

\frac{x + x + 3 0 ^{\circ}}{2} = \frac{12 x + 18 0 ^{\circ}}{12}

\frac{x + x + 3 0 ^{\circ}}{2}

Sumar elementos similares:

x + x = 2 x

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

Simplificar

2 x + 3 0^{\circ}

en una fracción:

\frac{12 x + 18 0 ^{\circ}}{6}

2 x + 3 0^{\circ}

Convertir a fracción:

2 x = \frac{2 x 6}{6}

= \frac{2 x \cdot 6}{6} + 3 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 x \cdot 6 + 18 0 ^{\circ}}{6}

Multiplicar los numeros:

2 \cdot 6 = 12

= \frac{12 x + 18 0 ^{\circ}}{6}

= \frac{\frac{12 x + 18 0 ^{\circ}}{6}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{12 x + 18 0 ^{\circ}}{6 \cdot 2}

Multiplicar los numeros:

6 \cdot 2 = 12

= \frac{12 x + 18 0 ^{\circ}}{12}

= - 2 sin (\frac{12 x + 18 0 ^{\circ}}{12}) sin (\frac{x - ( x + 3 0 ^{\circ} )}{2})

\frac{x - ( x + 3 0 ^{\circ} )}{2} = - 1 5^{\circ}

\frac{x - ( x + 3 0 ^{\circ} )}{2}

Expandir

x - (x + 3 0^{\circ}) : - 3 0^{\circ}

x - (x + 3 0^{\circ})

- (x + 3 0^{\circ}) : - x - 3 0^{\circ}

- (x + 3 0^{\circ})

Poner los parentesis

= - (x) - (3 0^{\circ})

Aplicar las reglas de los signos

+ (- a) = - a

= - x - 3 0^{\circ}

= x - x - 3 0^{\circ}

Sumar elementos similares:

x - x = 0

= - 3 0^{\circ}

= \frac{- 3 0 ^{\circ}}{2}

Aplicar las propiedades de las fracciones:

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

= - \frac{3 0 ^{\circ}}{2}

Aplicar las propiedades de las fracciones:

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

\frac{3 0 ^{\circ}}{2} = \frac{18 0 ^{\circ}}{6 \cdot 2}

= - \frac{18 0 ^{\circ}}{6 \cdot 2}

Multiplicar los numeros:

6 \cdot 2 = 12

= - 1 5^{\circ}

= - 2 sin (- 1 5^{\circ}) sin (\frac{12 x + 18 0 ^{\circ}}{12})

sin (- 1 5^{\circ}) = - \frac{6 - 2}{4}

sin (- 1 5^{\circ})

Utilizar la siguiente propiedad:

sin (- x) = - sin (x)

sin (- 1 5^{\circ}) = - sin (1 5^{\circ})

= - sin (1 5^{\circ})

Re-escribir usando identidades trigonométricas:

sin (1 5^{\circ}) = \frac{6 - 2}{4}

sin (1 5^{\circ})

Re-escribir usando identidades trigonométricas:

sin (4 5^{\circ}) cos (3 0^{\circ}) - cos (4 5^{\circ}) sin (3 0^{\circ})

sin (1 5^{\circ})

Escribir

sin (1 5^{\circ})

como

sin (4 5^{\circ} - 3 0^{\circ})

= sin (4 5^{\circ} - 3 0^{\circ})

Utilizar la identidad de diferencia de ángulos:

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

= sin (4 5^{\circ}) cos (3 0^{\circ}) - cos (4 5^{\circ}) sin (3 0^{\circ})

= sin (4 5^{\circ}) cos (3 0^{\circ}) - cos (4 5^{\circ}) sin (3 0^{\circ})

Utilizar la siguiente identidad trivial:

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

sin (4 5^{\circ})

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{2}{2}

Utilizar la siguiente identidad trivial:

cos (3 0^{\circ}) = \frac{3}{2}

cos (3 0^{\circ})

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{3}{2}

Utilizar la siguiente identidad trivial:

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

cos (4 5^{\circ})

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}

Utilizar la siguiente identidad trivial:

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 0^{\circ})

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{1}{2}

= \frac{2}{2} \cdot \frac{3}{2} - \frac{2}{2} \cdot \frac{1}{2}

Simplificar

\frac{2}{2} \cdot \frac{3}{2} - \frac{2}{2} \cdot \frac{1}{2} : \frac{6 - 2}{4}

\frac{2}{2} \cdot \frac{3}{2} - \frac{2}{2} \cdot \frac{1}{2}

\frac{2}{2} \cdot \frac{3}{2} = \frac{6}{4}

\frac{2}{2} \cdot \frac{3}{2}

Multiplicar fracciones:

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

= \frac{2 3}{2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{2 3}{4}

Simplificar

23 : 6

23

Aplicar las leyes de los exponentes:

a b = a \cdot b

23 = 2 \cdot 3

= 2 \cdot 3

Multiplicar los numeros:

2 \cdot 3 = 6

= 6

= \frac{6}{4}

\frac{2}{2} \cdot \frac{1}{2} = \frac{2}{4}

\frac{2}{2} \cdot \frac{1}{2}

Multiplicar fracciones:

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

= \frac{2 \cdot 1}{2 \cdot 2}

Simplificar

= \frac{2}{4}

= \frac{6}{4} - \frac{2}{4}

Aplicar la regla

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

= \frac{6 - 2}{4}

= \frac{6 - 2}{4}

= - \frac{6 - 2}{4}

= - 2 (- \frac{6 - 2}{4}) sin (\frac{12 x + 18 0 ^{\circ}}{12})

Aplicar la regla

- (- a) = a

= 2 sin (\frac{12 x + 18 0 ^{\circ}}{12}) \frac{6 - 2}{4}

Multiplicar fracciones:

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

= \frac{2 ( 6 - 2 )}{4} sin (\frac{12 x + 18 0 ^{\circ}}{12})

Eliminar los terminos comunes:

2

= \frac{6 - 2}{2} sin (\frac{12 x + 18 0 ^{\circ}}{12})

= \frac{6 - 2}{2} sin (\frac{12 x + 18 0 ^{\circ}}{12})

\frac{6 - 2}{2} sin (\frac{12 x + 18 0 ^{\circ}}{12}) = 0

Multiplicar ambos lados por

2

\frac{6 - 2}{2} sin (\frac{12 x + 18 0 ^{\circ}}{12}) = 0

Multiplicar ambos lados por

2

2 \cdot \frac{6 - 2}{2} sin (\frac{12 x + 18 0 ^{\circ}}{12}) = 0 \cdot 2

Simplificar

(6 - 2) sin (\frac{12 x + 18 0 ^{\circ}}{12}) = 0

(6 - 2) sin (\frac{12 x + 18 0 ^{\circ}}{12}) = 0

Dividir ambos lados entre

6 - 2

(6 - 2) sin (\frac{12 x + 18 0 ^{\circ}}{12}) = 0

Dividir ambos lados entre

6 - 2

\frac{( 6 - 2 ) sin ( \frac{12 x + 18 0 ^{\circ}}{12} )}{6 - 2} = \frac{0}{6 - 2}

Simplificar

sin (\frac{12 x + 18 0 ^{\circ}}{12}) = 0

sin (\frac{12 x + 18 0 ^{\circ}}{12}) = 0

Soluciones generales para

sin (\frac{12 x + 18 0 ^{\circ}}{12}) = 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{12 x + 18 0 ^{\circ}}{12} = 0 + 36 0^{\circ} n, \frac{12 x + 18 0 ^{\circ}}{12} = 18 0^{\circ} + 36 0^{\circ} n

\frac{12 x + 18 0 ^{\circ}}{12} = 0 + 36 0^{\circ} n, \frac{12 x + 18 0 ^{\circ}}{12} = 18 0^{\circ} + 36 0^{\circ} n

Resolver

\frac{12 x + 18 0 ^{\circ}}{12} = 0 + 36 0^{\circ} n : x = 36 0^{\circ} n - 1 5^{\circ}

\frac{12 x + 18 0 ^{\circ}}{12} = 0 + 36 0^{\circ} n

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

\frac{12 x + 18 0 ^{\circ}}{12} = 36 0^{\circ} n

Multiplicar ambos lados por

12

\frac{12 x + 18 0 ^{\circ}}{12} = 36 0^{\circ} n

Multiplicar ambos lados por

12

\frac{12 ( 12 x + 18 0 ^{\circ} )}{12} = 12 \cdot 36 0^{\circ} n

Simplificar

12 x + 18 0^{\circ} = 432 0^{\circ} n

12 x + 18 0^{\circ} = 432 0^{\circ} n

Mover

18 0^{\circ}

al lado derecho

12 x + 18 0^{\circ} = 432 0^{\circ} n

Restar

18 0^{\circ}

de ambos lados

12 x + 18 0^{\circ} - 18 0^{\circ} = 432 0^{\circ} n - 18 0^{\circ}

Simplificar

12 x = 432 0^{\circ} n - 18 0^{\circ}

12 x = 432 0^{\circ} n - 18 0^{\circ}

Dividir ambos lados entre

12

12 x = 432 0^{\circ} n - 18 0^{\circ}

Dividir ambos lados entre

12

\frac{12 x}{12} = \frac{432 0 ^{\circ} n}{12} - 1 5^{\circ}

Simplificar

x = 36 0^{\circ} n - 1 5^{\circ}

x = 36 0^{\circ} n - 1 5^{\circ}

Resolver

\frac{12 x + 18 0 ^{\circ}}{12} = 18 0^{\circ} + 36 0^{\circ} n : x = 16 5^{\circ} + 36 0^{\circ} n

\frac{12 x + 18 0 ^{\circ}}{12} = 18 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

12

\frac{12 x + 18 0 ^{\circ}}{12} = 18 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

12

\frac{12 ( 12 x + 18 0 ^{\circ} )}{12} = 216 0^{\circ} + 12 \cdot 36 0^{\circ} n

Simplificar

12 x + 18 0^{\circ} = 216 0^{\circ} + 432 0^{\circ} n

12 x + 18 0^{\circ} = 216 0^{\circ} + 432 0^{\circ} n

Mover

18 0^{\circ}

al lado derecho

12 x + 18 0^{\circ} = 216 0^{\circ} + 432 0^{\circ} n

Restar

18 0^{\circ}

de ambos lados

12 x + 18 0^{\circ} - 18 0^{\circ} = 216 0^{\circ} + 432 0^{\circ} n - 18 0^{\circ}

Simplificar

12 x = 198 0^{\circ} + 432 0^{\circ} n

12 x = 198 0^{\circ} + 432 0^{\circ} n

Dividir ambos lados entre

12

12 x = 198 0^{\circ} + 432 0^{\circ} n

Dividir ambos lados entre

12

\frac{12 x}{12} = 16 5^{\circ} + \frac{432 0 ^{\circ} n}{12}

Simplificar

x = 16 5^{\circ} + 36 0^{\circ} n

x = 16 5^{\circ} + 36 0^{\circ} n

x = 36 0^{\circ} n - 1 5^{\circ}, x = 16 5^{\circ} + 36 0^{\circ} n

Soluciones para el rango

0 \leq x < 18 0^{\circ}

x = 16 5^{\circ}

Combinar toda las soluciones

7 5^{\circ} or 16 5^{\circ}

Los intervalos entre ceros

0 < x < 7 5^{\circ}, 7 5^{\circ} < x < 16 5^{\circ}, 16 5^{\circ} < x < π

Resumir en una tabla:

cos (x) + cos (x + 3 0^{\circ}) cos (x) - cos (x + 3 0^{\circ}) (cos (x) + cos (x + 3 0^{\circ})) (cos (x) - cos (x + 3 0^{\circ})) x = 0 + + + 0 < x < 7 5^{\circ} + + + x = 7 5^{\circ} 0 + 0 7 5^{\circ} < x < 16 5^{\circ} - + - x = 16 5^{\circ} - 00 16 5^{\circ} < x < π - - + x = π - - +

Identificar los intervalos que cumplen la condición:

< 0

7 5^{\circ} < x < 16 5^{\circ}

Utilizar la periodicidad de

cos^{2} (x) - cos^{2} (x + 3 0^{\circ})

7 5^{\circ} + πn < x < 16 5^{\circ} + πn

Ejemplos populares

sin^2(x)+cos(x)>= 1

sin^{2} (x) + cos (x) \geq 1

cos(x)>= (sqrt(3))/2

cos (x) \geq \frac{3}{2}

cos(x)-1>= 0

cos (x) - 1 \geq 0

sin(x)>-2

sin (x) > - 2

solvefor x, pi/2-arctan(x)<0.0001 resolver para

x, \frac{π}{2} - arctan (x) < 0.0001