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Popular Trigonometria >

cos(x)=(2-tan(x))(1+sin(x))

Solução

cos (x) = (2 - tan (x)) (1 + sin (x))

Solução

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

Graus

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Passos da solução

cos (x) = (2 - tan (x)) (1 + sin (x))

Subtrair

(2 - tan (x)) (1 + sin (x))

de ambos os lados

cos (x) - (2 - tan (x)) (1 + sin (x)) = 0

Expresar com seno, cosseno

cos (x) - (1 + sin (x)) (2 - tan (x))

Utilizar a Identidade Básica da Trigonometria:

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

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

Simplificar

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

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

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

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

Simplificar

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

em uma fração:

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

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

Converter para fração:

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

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

Já que os denominadores são iguais, combinar as frações:

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

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

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

Multiplicar frações:

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

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

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

Converter para fração:

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

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

Já que os denominadores são iguais, combinar as frações:

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

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

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

cos (x) cos (x) - (2 cos (x) - sin (x)) (1 + sin (x))

cos (x) cos (x) = cos^{2} (x)

cos (x) cos (x)

Aplicar as propriedades dos expoentes:

a^{b} \cdot a^{c} = a^{b + c}

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

= cos^{1 + 1} (x)

Somar:

1 + 1 = 2

= cos^{2} (x)

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

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

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

\frac{cos ^{2} ( x ) - ( - sin ( x ) + 2 cos ( x ) ) ( 1 + sin ( x ) )}{cos ( x )} = 0

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

cos^{2} (x) - (- sin (x) + 2 cos (x)) (1 + sin (x)) = 0

Reeecreva usando identidades trigonométricas

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

Utilizar a identidade trigonométrica pitagórica:

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

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

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

Simplificar

1 - sin^{2} (x) - (- sin (x) + 2 cos (x)) (1 + sin (x)) : sin (x) - 2 cos (x) - 2 cos (x) sin (x) + 1

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

Expandir

- (- sin (x) + 2 cos (x)) (1 + sin (x)) : sin (x) + sin^{2} (x) - 2 cos (x) - 2 cos (x) sin (x)

Expandir

(- sin (x) + 2 cos (x)) (1 + sin (x)) : - sin (x) - sin^{2} (x) + 2 cos (x) + 2 cos (x) sin (x)

(- sin (x) + 2 cos (x)) (1 + sin (x))

Aplique o método FOIL:

(a + b) (c + d) = a c + a d + b c + b d

a = - sin (x), b = 2 cos (x), c = 1, d = sin (x)

= (- sin (x)) \cdot 1 + (- sin (x)) sin (x) + 2 cos (x) \cdot 1 + 2 cos (x) sin (x)

Aplicar as regras dos sinais

+ (- a) = - a

= - 1 \cdot sin (x) - sin (x) sin (x) + 2 \cdot 1 \cdot cos (x) + 2 cos (x) sin (x)

Simplificar

- 1 \cdot sin (x) - sin (x) sin (x) + 2 \cdot 1 \cdot cos (x) + 2 cos (x) sin (x) : - sin (x) - sin^{2} (x) + 2 cos (x) + 2 cos (x) sin (x)

- 1 \cdot sin (x) - sin (x) sin (x) + 2 \cdot 1 \cdot cos (x) + 2 cos (x) sin (x)

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

1 \cdot sin (x)

Multiplicar:

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

= sin (x)

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

Aplicar as propriedades dos expoentes:

a^{b} \cdot a^{c} = a^{b + c}

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

= sin^{1 + 1} (x)

Somar:

1 + 1 = 2

= sin^{2} (x)

2 \cdot 1 \cdot cos (x) = 2 cos (x)

2 \cdot 1 \cdot cos (x)

Multiplicar os números:

2 \cdot 1 = 2

= 2 cos (x)

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

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

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

Colocar os parênteses

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

Aplicar as regras dos sinais

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

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

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

Simplificar

1 - sin^{2} (x) + sin (x) + sin^{2} (x) - 2 cos (x) - 2 cos (x) sin (x) : sin (x) - 2 cos (x) - 2 cos (x) sin (x) + 1

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

Agrupar termos semelhantes

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

Somar elementos similares:

- sin^{2} (x) + sin^{2} (x) = 0

= sin (x) - 2 cos (x) - 2 cos (x) sin (x) + 1

= sin (x) - 2 cos (x) - 2 cos (x) sin (x) + 1

= sin (x) - 2 cos (x) - 2 cos (x) sin (x) + 1

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

Fatorar

1 + sin (x) - 2 cos (x) - 2 cos (x) sin (x) : (1 - 2 cos (x)) (sin (x) + 1)

1 + sin (x) - 2 cos (x) - 2 cos (x) sin (x)

Fatorar o termo comum

sin (x)

= 1 + sin (x) (1 - 2 cos (x)) - 2 cos (x)

Reescrever como

= (1 - 2 cos (x)) sin (x) + 1 \cdot (1 - 2 cos (x))

Fatorar o termo comum

(1 - 2 cos (x))

= (1 - 2 cos (x)) (sin (x) + 1)

(1 - 2 cos (x)) (sin (x) + 1) = 0

Resolver cada parte separadamente

1 - 2 cos (x) = 0 or sin (x) + 1 = 0

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

1 - 2 cos (x) = 0

Mova

1

para o lado direito

1 - 2 cos (x) = 0

Subtrair

1

de ambos os lados

1 - 2 cos (x) - 1 = 0 - 1

Simplificar

- 2 cos (x) = - 1

- 2 cos (x) = - 1

Dividir ambos os lados por

- 2

- 2 cos (x) = - 1

Dividir ambos os lados por

- 2

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

Simplificar

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

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

Soluções gerais para

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

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

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

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

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

sin (x) + 1 = 0

Mova

1

para o lado direito

sin (x) + 1 = 0

Subtrair

1

de ambos os lados

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

Simplificar

sin (x) = - 1

sin (x) = - 1

Soluções gerais para

sin (x) = - 1

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 = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

Combinar toda as soluções

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

Dado que a equação é indefinida para:

\frac{3 π}{2} + 2 πn

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

Gráfico

Exemplos populares

sin(θ)= 13/85

sin (θ) = \frac{13}{85}

sin(2x)=((6m-5))/8

sin (2 x) = \frac{( 6 m - 5 )}{8}

cos(θ)= 3/(sqrt(45))

cos (θ) = \frac{3}{45}

cos^2(x)= 9/16

cos^{2} (x) = \frac{9}{16}

sin(4x)=0.5

sin (4 x) = 0.5