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Popular Trigonometria >

(1-cot(x))(csc(x-pi/3)-2)<0,-pi<,x<0

Solução

(1 - cot (x)) (csc (x - \frac{π}{3}) - 2) < 0, - π <, x < 0

Solução

\frac{π}{4} + 2 πn < x < \frac{π}{3} + 2 πn or \frac{π}{2} + 2 πn < x < π + 2 πn or \frac{7 π}{6} + 2 πn < x < \frac{5 π}{4} + 2 πn or \frac{4 π}{3} + 2 πn < x < 2 π + 2 πn

Notação de intervalo

(\frac{π}{4} + 2 πn, \frac{π}{3} + 2 πn) \cup (\frac{π}{2} + 2 πn, π + 2 πn) \cup (\frac{7 π}{6} + 2 πn, \frac{5 π}{4} + 2 πn) \cup (\frac{4 π}{3} + 2 πn, 2 π + 2 πn)

Decimal

0.78539 \dots + 2 πn < x < 1.04719 \dots + 2 πn or 1.57079 \dots + 2 πn < x < 3.14159 \dots + 2 πn or 3.66519 \dots + 2 πn < x < 3.92699 \dots + 2 πn or 4.18879 \dots + 2 πn < x < 6.28318 \dots + 2 πn

Passos da solução

(1 - cot (x)) (csc (x - \frac{π}{3}) - 2) < 0

Periodicidade de

(1 - cot (x)) (csc (x - \frac{π}{3}) - 2) : 2 π

(1 - cot (x)) (csc (x - \frac{π}{3}) - 2)

é composta pelas seguintes funções e períodos:

cot (x)

com periodicidade de

π

A periodicidade composta é:

= 2 π

Expresar com seno, cosseno

(1 - cot (x)) (csc (x - \frac{π}{3}) - 2) < 0

Utilizar a Identidade Básica da Trigonometria:

cot (x) = \frac{cos ( x )}{sin ( x )}

(1 - \frac{cos ( x )}{sin ( x )}) (csc (x - \frac{π}{3}) - 2) < 0

Utilizar a Identidade Básica da Trigonometria:

csc (x) = \frac{1}{sin ( x )}

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

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

Simplificar

(1 - \frac{cos ( x )}{sin ( x )}) (\frac{1}{sin ( x - \frac{π}{3} )} - 2) : \frac{( 1 - 2 sin ( \frac{3 x - π}{3} ) ) ( sin ( x ) - cos ( x ) )}{sin ( \frac{3 x - π}{3} ) sin ( x )}

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

Simplificar

1 - \frac{cos ( x )}{sin ( x )}

em uma fração:

\frac{sin ( x ) - cos ( x )}{sin ( x )}

1 - \frac{cos ( x )}{sin ( x )}

Converter para fração:

1 = \frac{1 sin ( x )}{sin ( x )}

= \frac{1 \cdot sin ( x )}{sin ( x )} - \frac{cos ( x )}{sin ( x )}

Já que os denominadores são iguais, combinar as frações:

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

= \frac{1 \cdot sin ( x ) - cos ( x )}{sin ( x )}

Multiplicar:

1 \cdot sin (x) = sin (x)

= \frac{sin ( x ) - cos ( x )}{sin ( x )}

= \frac{sin ( x ) - cos ( x )}{sin ( x )} (\frac{1}{sin ( x - \frac{π}{3} )} - 2)

Simplificar

x - \frac{π}{3}

em uma fração:

\frac{3 x - π}{3}

x - \frac{π}{3}

Converter para fração:

x = \frac{x 3}{3}

= \frac{x \cdot 3}{3} - \frac{π}{3}

Já que os denominadores são iguais, combinar as frações:

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

= \frac{x \cdot 3 - π}{3}

= \frac{sin ( x ) - cos ( x )}{sin ( x )} (\frac{1}{sin ( \frac{3 x - π}{3} )} - 2)

Multiplicar frações:

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

= \frac{( sin ( x ) - cos ( x ) ) ( \frac{1}{s i n ( \frac{x \cdot 3 - π}{3} )} - 2 )}{sin ( x )}

Simplificar

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

em uma fração:

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

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

Converter para fração:

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

= \frac{1}{sin ( \frac{x \cdot 3 - π}{3} )} - \frac{2 sin ( \frac{x \cdot 3 - π}{3} )}{sin ( \frac{x \cdot 3 - π}{3} )}

Já que os denominadores são iguais, combinar as frações:

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

= \frac{1 - 2 sin ( \frac{x \cdot 3 - π}{3} )}{sin ( \frac{x \cdot 3 - π}{3} )}

= \frac{\frac{- 2 s i n ( \frac{3 x - π}{3} ) + 1}{s i n ( \frac{3 x - π}{3} )} ( sin ( x ) - cos ( x ) )}{sin ( x )}

Multiplicar

(sin (x) - cos (x)) \frac{1 - 2 sin ( \frac{x \cdot 3 - π}{3} )}{sin ( \frac{x \cdot 3 - π}{3} )} : \frac{( - 2 sin ( \frac{3 x - π}{3} ) + 1 ) ( sin ( x ) - cos ( x ) )}{sin ( \frac{x \cdot 3 - π}{3} )}

(sin (x) - cos (x)) \frac{1 - 2 sin ( \frac{x \cdot 3 - π}{3} )}{sin ( \frac{x \cdot 3 - π}{3} )}

Multiplicar frações:

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

= \frac{( 1 - 2 sin ( \frac{x \cdot 3 - π}{3} ) ) ( sin ( x ) - cos ( x ) )}{sin ( \frac{x \cdot 3 - π}{3} )}

= \frac{\frac{( - 2 s i n ( \frac{3 x - π}{3} ) + 1 ) ( s i n ( x ) - c o s ( x ) )}{s i n ( \frac{x \cdot 3 - π}{3} )}}{sin ( x )}

Aplicar as propriedades das frações:

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

= \frac{( 1 - 2 sin ( \frac{x \cdot 3 - π}{3} ) ) ( sin ( x ) - cos ( x ) )}{sin ( \frac{x \cdot 3 - π}{3} ) sin ( x )}

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

Encontre os zeros e pontos indefinidos de

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

para

0 \leq x < 2 π

Para encontrar os zeros, defina a desigualdade como zero

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

\frac{( 1 - 2 sin ( \frac{3 x - π}{3} ) ) ( sin ( x ) - cos ( x ) )}{sin ( \frac{3 x - π}{3} ) sin ( x )} = 0, 0 \leq x < 2 π : x = \frac{π}{2}, x = \frac{7 π}{6}, x = \frac{π}{4}, x = \frac{5 π}{4}

\frac{( 1 - 2 sin ( \frac{3 x - π}{3} ) ) ( sin ( x ) - cos ( x ) )}{sin ( \frac{3 x - π}{3} ) sin ( x )} = 0, 0 \leq x < 2 π

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Resolver cada parte separadamente

1 - 2 sin (\frac{3 x - π}{3}) = 0 or sin (x) - cos (x) = 0

1 - 2 sin (\frac{3 x - π}{3}) = 0, 0 \leq x < 2 π : x = \frac{π}{2}, x = \frac{7 π}{6}

1 - 2 sin (\frac{3 x - π}{3}) = 0, 0 \leq x < 2 π

Mova

1

para o lado direito

1 - 2 sin (\frac{3 x - π}{3}) = 0

Subtrair

1

de ambos os lados

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

Simplificar

- 2 sin (\frac{3 x - π}{3}) = - 1

- 2 sin (\frac{3 x - π}{3}) = - 1

Dividir ambos os lados por

- 2

- 2 sin (\frac{3 x - π}{3}) = - 1

Dividir ambos os lados por

- 2

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

Simplificar

sin (\frac{3 x - π}{3}) = \frac{1}{2}

sin (\frac{3 x - π}{3}) = \frac{1}{2}

Soluções gerais para

sin (\frac{3 x - π}{3}) = \frac{1}{2}

sin (x)

tabela de periodicidade com ciclo de

2 πn

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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} 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 - π}{3} = \frac{π}{6} + 2 πn, \frac{3 x - π}{3} = \frac{5 π}{6} + 2 πn

\frac{3 x - π}{3} = \frac{π}{6} + 2 πn, \frac{3 x - π}{3} = \frac{5 π}{6} + 2 πn

Resolver

\frac{3 x - π}{3} = \frac{π}{6} + 2 πn : x = 2 πn + \frac{π}{3} + \frac{π}{6}

\frac{3 x - π}{3} = \frac{π}{6} + 2 πn

Multiplicar ambos os lados por

3

\frac{3 x - π}{3} = \frac{π}{6} + 2 πn

Multiplicar ambos os lados por

3

\frac{3 ( 3 x - π )}{3} = 3 \cdot \frac{π}{6} + 3 \cdot 2 πn

Simplificar

\frac{3 ( 3 x - π )}{3} = 3 \cdot \frac{π}{6} + 3 \cdot 2 πn

Simplificar

\frac{3 ( 3 x - π )}{3} : 3 x - π

\frac{3 ( 3 x - π )}{3}

Dividir:

\frac{3}{3} = 1

= 3 x - π

Simplificar

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

3 \cdot \frac{π}{6} + 3 \cdot 2 πn

3 \cdot \frac{π}{6} = \frac{π}{2}

3 \cdot \frac{π}{6}

Multiplicar frações:

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

= \frac{π 3}{6}

Eliminar o fator comum:

3

= \frac{π}{2}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiplicar os números:

3 \cdot 2 = 6

= 6 πn

= \frac{π}{2} + 6 πn

3 x - π = \frac{π}{2} + 6 πn

3 x - π = \frac{π}{2} + 6 πn

3 x - π = \frac{π}{2} + 6 πn

Mova

π

para o lado direito

3 x - π = \frac{π}{2} + 6 πn

Adicionar

π

a ambos os lados

3 x - π + π = \frac{π}{2} + 6 πn + π

Simplificar

3 x = \frac{π}{2} + 6 πn + π

3 x = \frac{π}{2} + 6 πn + π

Dividir ambos os lados por

3

3 x = \frac{π}{2} + 6 πn + π

Dividir ambos os lados por

3

\frac{3 x}{3} = \frac{\frac{π}{2}}{3} + \frac{6 πn}{3} + \frac{π}{3}

Simplificar

\frac{3 x}{3} = \frac{\frac{π}{2}}{3} + \frac{6 πn}{3} + \frac{π}{3}

Simplificar

\frac{3 x}{3} : x

\frac{3 x}{3}

Dividir:

\frac{3}{3} = 1

= x

Simplificar

\frac{\frac{π}{2}}{3} + \frac{6 πn}{3} + \frac{π}{3} : 2 πn + \frac{π}{3} + \frac{π}{6}

\frac{\frac{π}{2}}{3} + \frac{6 πn}{3} + \frac{π}{3}

Agrupar termos semelhantes

= \frac{π}{3} + \frac{6 πn}{3} + \frac{\frac{π}{2}}{3}

\frac{6 πn}{3} = 2 πn

\frac{6 πn}{3}

Dividir:

\frac{6}{3} = 2

= 2 πn

\frac{\frac{π}{2}}{3} = \frac{π}{6}

\frac{\frac{π}{2}}{3}

Aplicar as propriedades das frações:

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

= \frac{π}{2 \cdot 3}

Multiplicar os números:

2 \cdot 3 = 6

= \frac{π}{6}

= \frac{π}{3} + 2 πn + \frac{π}{6}

Agrupar termos semelhantes

= 2 πn + \frac{π}{3} + \frac{π}{6}

x = 2 πn + \frac{π}{3} + \frac{π}{6}

x = 2 πn + \frac{π}{3} + \frac{π}{6}

x = 2 πn + \frac{π}{3} + \frac{π}{6}

Resolver

\frac{3 x - π}{3} = \frac{5 π}{6} + 2 πn : x = 2 πn + \frac{π}{3} + \frac{5 π}{6}

\frac{3 x - π}{3} = \frac{5 π}{6} + 2 πn

Multiplicar ambos os lados por

3

\frac{3 x - π}{3} = \frac{5 π}{6} + 2 πn

Multiplicar ambos os lados por

3

\frac{3 ( 3 x - π )}{3} = 3 \cdot \frac{5 π}{6} + 3 \cdot 2 πn

Simplificar

\frac{3 ( 3 x - π )}{3} = 3 \cdot \frac{5 π}{6} + 3 \cdot 2 πn

Simplificar

\frac{3 ( 3 x - π )}{3} : 3 x - π

\frac{3 ( 3 x - π )}{3}

Dividir:

\frac{3}{3} = 1

= 3 x - π

Simplificar

3 \cdot \frac{5 π}{6} + 3 \cdot 2 πn : \frac{5 π}{2} + 6 πn

3 \cdot \frac{5 π}{6} + 3 \cdot 2 πn

3 \cdot \frac{5 π}{6} = \frac{5 π}{2}

3 \cdot \frac{5 π}{6}

Multiplicar frações:

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

= \frac{5 π 3}{6}

Multiplicar os números:

5 \cdot 3 = 15

= \frac{15 π}{6}

Eliminar o fator comum:

3

= \frac{5 π}{2}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiplicar os números:

3 \cdot 2 = 6

= 6 πn

= \frac{5 π}{2} + 6 πn

3 x - π = \frac{5 π}{2} + 6 πn

3 x - π = \frac{5 π}{2} + 6 πn

3 x - π = \frac{5 π}{2} + 6 πn

Mova

π

para o lado direito

3 x - π = \frac{5 π}{2} + 6 πn

Adicionar

π

a ambos os lados

3 x - π + π = \frac{5 π}{2} + 6 πn + π

Simplificar

3 x = \frac{5 π}{2} + 6 πn + π

3 x = \frac{5 π}{2} + 6 πn + π

Dividir ambos os lados por

3

3 x = \frac{5 π}{2} + 6 πn + π

Dividir ambos os lados por

3

\frac{3 x}{3} = \frac{\frac{5 π}{2}}{3} + \frac{6 πn}{3} + \frac{π}{3}

Simplificar

\frac{3 x}{3} = \frac{\frac{5 π}{2}}{3} + \frac{6 πn}{3} + \frac{π}{3}

Simplificar

\frac{3 x}{3} : x

\frac{3 x}{3}

Dividir:

\frac{3}{3} = 1

= x

Simplificar

\frac{\frac{5 π}{2}}{3} + \frac{6 πn}{3} + \frac{π}{3} : 2 πn + \frac{π}{3} + \frac{5 π}{6}

\frac{\frac{5 π}{2}}{3} + \frac{6 πn}{3} + \frac{π}{3}

Agrupar termos semelhantes

= \frac{π}{3} + \frac{6 πn}{3} + \frac{\frac{5 π}{2}}{3}

\frac{6 πn}{3} = 2 πn

\frac{6 πn}{3}

Dividir:

\frac{6}{3} = 2

= 2 πn

\frac{\frac{5 π}{2}}{3} = \frac{5 π}{6}

\frac{\frac{5 π}{2}}{3}

Aplicar as propriedades das frações:

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

= \frac{5 π}{2 \cdot 3}

Multiplicar os números:

2 \cdot 3 = 6

= \frac{5 π}{6}

= \frac{π}{3} + 2 πn + \frac{5 π}{6}

Agrupar termos semelhantes

= 2 πn + \frac{π}{3} + \frac{5 π}{6}

x = 2 πn + \frac{π}{3} + \frac{5 π}{6}

x = 2 πn + \frac{π}{3} + \frac{5 π}{6}

x = 2 πn + \frac{π}{3} + \frac{5 π}{6}

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

Soluções para o intervalo

0 \leq x < 2 π

x = \frac{π}{2}, x = \frac{7 π}{6}

sin (x) - cos (x) = 0, 0 \leq x < 2 π : x = \frac{π}{4}, x = \frac{5 π}{4}

sin (x) - cos (x) = 0, 0 \leq x < 2 π

Reeecreva usando identidades trigonométricas

sin (x) - cos (x) = 0

Dividir ambos os lados por

cos (x), cos (x) \neq = 0

\frac{sin ( x ) - cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplificar

\frac{sin ( x )}{cos ( x )} - 1 = 0

Utilizar a Identidade Básica da Trigonometria:

\frac{sin ( x )}{cos ( x )} = tan (x)

tan (x) - 1 = 0

tan (x) - 1 = 0

Mova

1

para o lado direito

tan (x) - 1 = 0

Adicionar

1

a ambos os lados

tan (x) - 1 + 1 = 0 + 1

Simplificar

tan (x) = 1

tan (x) = 1

Soluções gerais para

tan (x) = 1

tan (x)

tabela de periodicidade com ciclo de

πn

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Soluções para o intervalo

0 \leq x < 2 π

x = \frac{π}{4}, x = \frac{5 π}{4}

Combinar toda as soluções

x = \frac{π}{2}, x = \frac{7 π}{6}, x = \frac{π}{4}, x = \frac{5 π}{4}

Encontre os pontos indefinidos:

x = \frac{π}{3}, x = \frac{4 π}{3}, x = 0, x = π

Encontre os zeros do denominador

sin (\frac{3 x - π}{3}) sin (x) = 0

Resolver cada parte separadamente

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

sin (\frac{3 x - π}{3}) = 0, 0 \leq x < 2 π : x = \frac{π}{3}, x = \frac{4 π}{3}

sin (\frac{3 x - π}{3}) = 0, 0 \leq x < 2 π

Soluções gerais para

sin (\frac{3 x - π}{3}) = 0

sin (x)

tabela de periodicidade com ciclo de

2 πn

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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} 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 - π}{3} = 0 + 2 πn, \frac{3 x - π}{3} = π + 2 πn

\frac{3 x - π}{3} = 0 + 2 πn, \frac{3 x - π}{3} = π + 2 πn

Resolver

\frac{3 x - π}{3} = 0 + 2 πn : x = 2 πn + \frac{π}{3}

\frac{3 x - π}{3} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{3 x - π}{3} = 2 πn

Multiplicar ambos os lados por

3

\frac{3 x - π}{3} = 2 πn

Multiplicar ambos os lados por

3

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

Simplificar

3 x - π = 6 πn

3 x - π = 6 πn

Mova

π

para o lado direito

3 x - π = 6 πn

Adicionar

π

a ambos os lados

3 x - π + π = 6 πn + π

Simplificar

3 x = 6 πn + π

3 x = 6 πn + π

Dividir ambos os lados por

3

3 x = 6 πn + π

Dividir ambos os lados por

3

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

Simplificar

x = 2 πn + \frac{π}{3}

x = 2 πn + \frac{π}{3}

Resolver

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

\frac{3 x - π}{3} = π + 2 πn

Multiplicar ambos os lados por

3

\frac{3 x - π}{3} = π + 2 πn

Multiplicar ambos os lados por

3

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

Simplificar

3 x - π = 3 π + 6 πn

3 x - π = 3 π + 6 πn

Mova

π

para o lado direito

3 x - π = 3 π + 6 πn

Adicionar

π

a ambos os lados

3 x - π + π = 3 π + 6 πn + π

Simplificar

3 x = 4 π + 6 πn

3 x = 4 π + 6 πn

Dividir ambos os lados por

3

3 x = 4 π + 6 πn

Dividir ambos os lados por

3

\frac{3 x}{3} = \frac{4 π}{3} + \frac{6 πn}{3}

Simplificar

x = \frac{4 π}{3} + 2 πn

x = \frac{4 π}{3} + 2 πn

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

Soluções para o intervalo

0 \leq x < 2 π

x = \frac{π}{3}, x = \frac{4 π}{3}

sin (x) = 0, 0 \leq x < 2 π : x = 0, x = π

sin (x) = 0, 0 \leq x < 2 π

Soluções gerais para

sin (x) = 0

sin (x)

tabela de periodicidade com ciclo de

2 πn

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

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

Soluções para o intervalo

0 \leq x < 2 π

x = 0, x = π

Combinar toda as soluções

x = \frac{π}{3}, x = \frac{4 π}{3}, x = 0, x = π

0, \frac{π}{4}, \frac{π}{3}, \frac{π}{2}, π, \frac{7 π}{6}, \frac{5 π}{4}, \frac{4 π}{3}

Identifique os intervalos

0 < x < \frac{π}{4}, \frac{π}{4} < x < \frac{π}{3}, \frac{π}{3} < x < \frac{π}{2}, \frac{π}{2} < x < π, π < x < \frac{7 π}{6}, \frac{7 π}{6} < x < \frac{5 π}{4}, \frac{5 π}{4} < x < \frac{4 π}{3}, \frac{4 π}{3} < x < 2 π

Resumir em uma tabela:

1 - 2 sin (\frac{3 x - π}{3}) sin (x) - cos (x) sin (\frac{3 x - π}{3}) sin (x) \frac{( 1 - 2 s i n ( \frac{3 x - π}{3} ) ) ( s i n ( x ) - c o s ( x ) )}{s i n ( \frac{3 x - π}{3} ) s i n ( x )} x = 0 + - - 0 Indefinido 0 < x < \frac{π}{4} + - - + + x = \frac{π}{4} + 0 - + 0 \frac{π}{4} < x < \frac{π}{3} + + - + - x = \frac{π}{3} + + 0 + Indefinido \frac{π}{3} < x < \frac{π}{2} + + + + + x = \frac{π}{2} 0 + + + 0 \frac{π}{2} < x < π - + + + - x = π - + + 0 Indefinido π < x < \frac{7 π}{6} - + + - + x = \frac{7 π}{6} 0 + + - 0 \frac{7 π}{6} < x < \frac{5 π}{4} + + + - - x = \frac{5 π}{4} + 0 + - 0 \frac{5 π}{4} < x < \frac{4 π}{3} + - + - + x = \frac{4 π}{3} + - 0 - Indefinido \frac{4 π}{3} < x < 2 π + - - - - x = 2 π + - - 0 Indefinido

Identifique os intervalos que satisfaçam à condição necessária:

< 0

\frac{π}{4} < x < \frac{π}{3} or \frac{π}{2} < x < π or \frac{7 π}{6} < x < \frac{5 π}{4} or \frac{4 π}{3} < x < 2 π

Utilizar a periodicidade de

(1 - cot (x)) (csc (x - \frac{π}{3}) - 2)

\frac{π}{4} + 2 πn < x < \frac{π}{3} + 2 πn or \frac{π}{2} + 2 πn < x < π + 2 πn or \frac{7 π}{6} + 2 πn < x < \frac{5 π}{4} + 2 πn or \frac{4 π}{3} + 2 πn < x < 2 π + 2 πn

Exemplos populares

(cos(x))^2>= 1

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

sin(x)<= (sqrt(3))/2 ,-pi<= x<= pi

sin (x) \leq \frac{3}{2}, - π \leq x \leq π

2cos(4x-pi/3)-1<= 0

2 cos (4 x - \frac{π}{3}) - 1 \leq 0

sin(x)<=-5/2

sin (x) \leq - \frac{5}{2}

arccos((x-5)/(10))-pi/3 >= 0

arccos (\frac{x - 5}{10}) - \frac{π}{3} \geq 0