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Popular Trigonometria >

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

Solução

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

Solução

x = 2 πn, x = \frac{π}{4} + πn

Graus

x = 0^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

Passos da solução

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

Subtrair

cos (x) - 1

de ambos os lados

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

Expresar com seno, cosseno

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

Utilizar a Identidade Básica da Trigonometria:

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

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

Simplificar

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

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

Converter para fração:

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

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

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

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

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

1 \cdot cos (x)

Multiplicar:

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

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

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

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

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

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

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

Fatorar

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

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

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

Fatorar

sin (x)

sin (x) cos (x) - sin (x)

sin (x) (cos (x) - 1)

sin (x) cos (x) - sin (x)

Fatorar o termo comum

sin (x)

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

Fatorar

- cos (x)

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

- cos (x) (cos (x) - 1)

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

Aplicar as propriedades dos expoentes:

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

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

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

Fatorar o termo comum

- cos (x)

= - cos (x) (cos (x) - 1)

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

Fatorar o termo comum

cos (x) - 1

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

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

Resolver cada parte separadamente

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

cos (x) - 1 = 0 : x = 2 πn

cos (x) - 1 = 0

Mova

1

para o lado direito

cos (x) - 1 = 0

Adicionar

1

a ambos os lados

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

Simplificar

cos (x) = 1

cos (x) = 1

Soluções gerais para

cos (x) = 1

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 = 0 + 2 πn

x = 0 + 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

sin (x) - cos (x) = 0 : x = \frac{π}{4} + πn

sin (x) - cos (x) = 0

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

Combinar toda as soluções

x = 2 πn, x = \frac{π}{4} + πn

Gráfico

Exemplos populares

2-8cos^2(x)=0

2 - 8 cos^{2} (x) = 0

sin(2θ-pi/2)=1

sin (2 θ - \frac{π}{2}) = 1

2sin^2(2θ)=1-sin(2θ)

2 sin^{2} (2 θ) = 1 - sin (2 θ)

cos(θ)= 24/25

cos (θ) = \frac{24}{25}

sec^2(x)=1-tan^2(x)

sec^{2} (x) = 1 - tan^{2} (x)