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Popular Trigonometria >

(cos(-30+x))/(cos(30+x))= 1835/726

Solução

\frac{cos ( - 3 0 ^{\circ} + x )}{cos ( 3 0 ^{\circ} + x )} = \frac{1835}{726}

Solução

x = 0.64352 \dots + 18 0^{\circ} n

Radianos

x = 0.64352 \dots + πn

Passos da solução

\frac{cos ( - 3 0 ^{\circ} + x )}{cos ( 3 0 ^{\circ} + x )} = \frac{1835}{726}

Reeecreva usando identidades trigonométricas

\frac{cos ( - 3 0 ^{\circ} + x )}{cos ( 3 0 ^{\circ} + x )} = \frac{1835}{726}

Reeecreva usando identidades trigonométricas

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

Use a identidade de soma de ângulos:

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

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

Simplificar

cos (3 0^{\circ}) cos (x) - sin (3 0^{\circ}) sin (x) : \frac{3}{2} cos (x) - \frac{1}{2} sin (x)

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

Simplificar

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

cos (3 0^{\circ})

Utilizar a seguinte identidade trivial:

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

cos (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

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}

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

Simplificar

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

sin (3 0^{\circ})

Utilizar a seguinte identidade trivial:

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

sin (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

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

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

Use a identidade de diferença de ângulos:

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

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

Simplificar

cos (x) cos (3 0^{\circ}) + sin (x) sin (3 0^{\circ}) : \frac{3}{2} cos (x) + \frac{1}{2} sin (x)

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

Simplificar

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

cos (3 0^{\circ})

Utilizar a seguinte identidade trivial:

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

cos (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

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}

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

Simplificar

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

sin (3 0^{\circ})

Utilizar a seguinte identidade trivial:

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

sin (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

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

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

\frac{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )}{\frac{3}{2} cos ( x ) - \frac{1}{2} sin ( x )} = \frac{1835}{726}

\frac{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )}{\frac{3}{2} cos ( x ) - \frac{1}{2} sin ( x )} = \frac{1835}{726}

Subtrair

\frac{1835}{726}

de ambos os lados

\frac{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )}{\frac{3}{2} cos ( x ) - \frac{1}{2} sin ( x )} - \frac{1835}{726} = 0

Simplificar

\frac{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )}{\frac{3}{2} cos ( x ) - \frac{1}{2} sin ( x )} - \frac{1835}{726} : \frac{- 1109 3 cos ( x ) + 2561 sin ( x )}{726 ( 3 cos ( x ) - sin ( x ) )}

\frac{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )}{\frac{3}{2} cos ( x ) - \frac{1}{2} sin ( x )} - \frac{1835}{726}

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

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

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

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

Multiplicar frações:

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

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

\frac{1}{2} sin (x) = \frac{sin ( x )}{2}

\frac{1}{2} sin (x)

Multiplicar frações:

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

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

Multiplicar:

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

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

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

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

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

Multiplicar frações:

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

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

\frac{1}{2} sin (x) = \frac{sin ( x )}{2}

\frac{1}{2} sin (x)

Multiplicar frações:

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

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

Multiplicar:

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

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

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

Combinar as frações usando o mínimo múltiplo comum:

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

Aplicar a regra

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

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

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

Combinar as frações usando o mínimo múltiplo comum:

\frac{3 cos ( x ) + sin ( x )}{2}

Aplicar a regra

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

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

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

Dividir frações:

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

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

Eliminar o fator comum:

2

= \frac{3 cos ( x ) + sin ( x )}{3 cos ( x ) - sin ( x )}

= \frac{3 cos ( x ) + sin ( x )}{3 cos ( x ) - sin ( x )} - \frac{1835}{726}

Mínimo múltiplo comum de

3 cos (x) - sin (x), 726 : 726 (3 cos (x) - sin (x))

3 cos (x) - sin (x), 726

Mínimo múltiplo comum (MMC)

Calcular uma expressão que seja composta por fatores que estejam presentes tanto em

3 cos (x) - sin (x)

quanto em

726

= 726 (3 cos (x) - sin (x))

Reescrever as frações baseando-se no mínimo múltiplo comum

Multiplicar cada numerador pelo mesmo valor necessário para multiplicar seu denominador correspondente para convertê-lo no mínimo múltiplo comum

Para

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

multiplique o numerador e o denominador por

726

\frac{3 cos ( x ) + sin ( x )}{3 cos ( x ) - sin ( x )} = \frac{( 3 cos ( x ) + sin ( x ) ) \cdot 726}{( 3 cos ( x ) - sin ( x ) ) \cdot 726}

Para

\frac{1835}{726} :

multiplique o numerador e o denominador por

3 cos (x) - sin (x)

\frac{1835}{726} = \frac{1835 ( 3 cos ( x ) - sin ( x ) )}{726 ( 3 cos ( x ) - sin ( x ) )}

= \frac{( 3 cos ( x ) + sin ( x ) ) \cdot 726}{( 3 cos ( x ) - sin ( x ) ) \cdot 726} - \frac{1835 ( 3 cos ( x ) - sin ( x ) )}{726 ( 3 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{( 3 cos ( x ) + sin ( x ) ) \cdot 726 - 1835 ( 3 cos ( x ) - sin ( x ) )}{726 ( 3 cos ( x ) - sin ( x ) )}

Expandir

(3 cos (x) + sin (x)) \cdot 726 - 1835 (3 cos (x) - sin (x)) : - 11093 cos (x) + 2561 sin (x)

(3 cos (x) + sin (x)) \cdot 726 - 1835 (3 cos (x) - sin (x))

= 726 (3 cos (x) + sin (x)) - 1835 (3 cos (x) - sin (x))

Expandir

726 (3 cos (x) + sin (x)) : 7263 cos (x) + 726 sin (x)

726 (3 cos (x) + sin (x))

Colocar os parênteses utilizando:

a (b + c) = ab + a c

a = 726, b = 3 cos (x), c = sin (x)

= 7263 cos (x) + 726 sin (x)

= 7263 cos (x) + 726 sin (x) - 1835 (3 cos (x) - sin (x))

Expandir

- 1835 (3 cos (x) - sin (x)) : - 18353 cos (x) + 1835 sin (x)

- 1835 (3 cos (x) - sin (x))

Colocar os parênteses utilizando:

a (b - c) = ab - a c

a = - 1835, b = 3 cos (x), c = sin (x)

= - 18353 cos (x) - (- 1835) sin (x)

Aplicar as regras dos sinais

- (- a) = a

= - 18353 cos (x) + 1835 sin (x)

= 7263 cos (x) + 726 sin (x) - 18353 cos (x) + 1835 sin (x)

Simplificar

7263 cos (x) + 726 sin (x) - 18353 cos (x) + 1835 sin (x) : - 11093 cos (x) + 2561 sin (x)

7263 cos (x) + 726 sin (x) - 18353 cos (x) + 1835 sin (x)

Somar elementos similares:

7263 cos (x) - 18353 cos (x) = - 11093 cos (x)

= - 11093 cos (x) + 726 sin (x) + 1835 sin (x)

Somar elementos similares:

726 sin (x) + 1835 sin (x) = 2561 sin (x)

= - 11093 cos (x) + 2561 sin (x)

= - 11093 cos (x) + 2561 sin (x)

= \frac{- 1109 3 cos ( x ) + 2561 sin ( x )}{726 ( 3 cos ( x ) - sin ( x ) )}

\frac{- 1109 3 cos ( x ) + 2561 sin ( x )}{726 ( 3 cos ( x ) - sin ( x ) )} = 0

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

- 11093 cos (x) + 2561 sin (x) = 0

Reeecreva usando identidades trigonométricas

- 11093 cos (x) + 2561 sin (x) = 0

Dividir ambos os lados por

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

\frac{- 1109 3 cos ( x ) + 2561 sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplificar

- 11093 + \frac{2561 sin ( x )}{cos ( x )} = 0

Utilizar a Identidade Básica da Trigonometria:

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

- 11093 + 2561 tan (x) = 0

- 11093 + 2561 tan (x) = 0

Mova

11093

para o lado direito

- 11093 + 2561 tan (x) = 0

Adicionar

11093

a ambos os lados

- 11093 + 2561 tan (x) + 11093 = 0 + 11093

Simplificar

2561 tan (x) = 11093

2561 tan (x) = 11093

Dividir ambos os lados por

2561

2561 tan (x) = 11093

Dividir ambos os lados por

2561

\frac{2561 tan ( x )}{2561} = \frac{1109 3}{2561}

Simplificar

tan (x) = \frac{1109 3}{2561}

tan (x) = \frac{1109 3}{2561}

Aplique as propriedades trigonométricas inversas

tan (x) = \frac{1109 3}{2561}

Soluções gerais para

tan (x) = \frac{1109 3}{2561}

tan (x) = a \Rightarrow x = arctan (a) + 18 0^{\circ} n

x = arctan (\frac{1109 3}{2561}) + 18 0^{\circ} n

x = arctan (\frac{1109 3}{2561}) + 18 0^{\circ} n

Mostrar soluções na forma decimal

x = 0.64352 \dots + 18 0^{\circ} n

Gráfico

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