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Popular Trigonometria >

(sec(x)sin(x))/(tan(x)cot(x))=sin^2(x)

Solução

\frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} = sin^{2} (x)

Solução

Sem solu \overset{c}{\c} \tilde{a} o para x \in R

Passos da solução

\frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} = sin^{2} (x)

Subtrair

sin^{2} (x)

de ambos os lados

\frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} - sin^{2} (x) = 0

Simplificar

\frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} - sin^{2} (x) : \frac{sec ( x ) sin ( x ) - sin ^{2} ( x ) tan ( x ) cot ( x )}{tan ( x ) cot ( x )}

\frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} - sin^{2} (x)

Converter para fração:

sin^{2} (x) = \frac{sin ^{2} ( x ) tan ( x ) cot ( x )}{tan ( x ) cot ( x )}

= \frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} - \frac{sin ^{2} ( x ) tan ( x ) cot ( x )}{tan ( x ) cot ( x )}

Já que os denominadores são iguais, combinar as frações:

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

= \frac{sec ( x ) sin ( x ) - sin ^{2} ( x ) tan ( x ) cot ( x )}{tan ( x ) cot ( x )}

\frac{sec ( x ) sin ( x ) - sin ^{2} ( x ) tan ( x ) cot ( x )}{tan ( x ) cot ( x )} = 0

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

sec (x) sin (x) - sin^{2} (x) tan (x) cot (x) = 0

Fatorar

sec (x) sin (x) - sin^{2} (x) tan (x) cot (x) : sin (x) (sec (x) - sin (x) tan (x) cot (x))

sec (x) sin (x) - sin^{2} (x) tan (x) cot (x)

Aplicar as propriedades dos expoentes:

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

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

= sec (x) sin (x) - sin (x) sin (x) tan (x) cot (x)

Fatorar o termo comum

sin (x)

= sin (x) (sec (x) - sin (x) tan (x) cot (x))

sin (x) (sec (x) - sin (x) tan (x) cot (x)) = 0

Resolver cada parte separadamente

sin (x) = 0 or sec (x) - sin (x) tan (x) cot (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

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

sec (x) - sin (x) tan (x) cot (x) = 0 :

Sem solução

sec (x) - sin (x) tan (x) cot (x) = 0

Expresar com seno, cosseno

sec (x) - cot (x) sin (x) tan (x)

Utilizar a Identidade Básica da Trigonometria:

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

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

Utilizar a Identidade Básica da Trigonometria:

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

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

Utilizar a Identidade Básica da Trigonometria:

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

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

Simplificar

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

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

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

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

Multiplicar frações:

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

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

Eliminar o fator comum:

cos (x)

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

Eliminar o fator comum:

sin (x)

= sin (x)

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

Converter para fração:

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

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

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

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

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

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

Reeecreva usando identidades trigonométricas

1 - cos (x) sin (x)

Utilizar a identidade trigonométrica do arco duplo:

2 sin (x) cos (x) = sin (2 x)

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

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

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

Mova

1

para o lado direito

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

Subtrair

1

de ambos os lados

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

Simplificar

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

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

Multiplicar ambos os lados por

2

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

Multiplicar ambos os lados por

2

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

Simplificar

- sin (2 x) = - 2

- sin (2 x) = - 2

Dividir ambos os lados por

- 1

- sin (2 x) = - 2

Dividir ambos os lados por

- 1

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

Simplificar

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

Sem solu \overset{c}{\c} \tilde{a} o

Combinar toda as soluções

x = 2 πn, x = π + 2 πn

Dado que a equação é indefinida para:

2 πn, π + 2 πn

Sem solu \overset{c}{\c} \tilde{a} o para x \in R

Gráfico

Exemplos populares

sin(3x)+cos(3x)=0

sin (3 x) + cos (3 x) = 0

sin(x)=-3/4 ,pi<x<(3pi)/4

sin (x) = - \frac{3}{4}, π < x < \frac{3 π}{4}

cos(2x)+cos(x)=0,0<= x<2pi

cos (2 x) + cos (x) = 0, 0 \leq x < 2 π

10cos^2(x)=5

10 cos^{2} (x) = 5

cos(θ)= 5/9

cos (θ) = \frac{5}{9}