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Popular Trigonometria >

sin(θ)=0.6367+0.1*cos(θ)

Solução

sin (θ) = 0.6367 + 0.1 \cdot cos (θ)

Solução

θ = 2.55514 \dots + 2 πn, θ = 0.78578 \dots + 2 πn

Graus

θ = 146.39879 \dots^{\circ} + 36 0^{\circ} n, θ = 45.02238 \dots^{\circ} + 36 0^{\circ} n

Passos da solução

sin (θ) = 0.6367 + 0.1 cos (θ)

Elevar ambos os lados ao quadrado

sin^{2} (θ) = (0.6367 + 0.1 cos (θ))^{2}

Subtrair

(0.6367 + 0.1 cos (θ))^{2}

de ambos os lados

sin^{2} (θ) - 0.40538689 - 0.12734 cos (θ) - 0.01 cos^{2} (θ) = 0

Reeecreva usando identidades trigonométricas

- 0.40538689 + sin^{2} (θ) - 0.01 cos^{2} (θ) - 0.12734 cos (θ)

Utilizar a identidade trigonométrica pitagórica:

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

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

= - 0.40538689 + 1 - cos^{2} (θ) - 0.01 cos^{2} (θ) - 0.12734 cos (θ)

Simplificar

- 0.40538689 + 1 - cos^{2} (θ) - 0.01 cos^{2} (θ) - 0.12734 cos (θ) : - 1.01 cos^{2} (θ) - 0.12734 cos (θ) + 0.59461311

- 0.40538689 + 1 - cos^{2} (θ) - 0.01 cos^{2} (θ) - 0.12734 cos (θ)

Somar elementos similares:

- cos^{2} (θ) - 0.01 cos^{2} (θ) = - 1.01 cos^{2} (θ)

= - 0.40538689 + 1 - 1.01 cos^{2} (θ) - 0.12734 cos (θ)

Somar/subtrair:

- 0.40538689 + 1 = 0.59461311

= - 1.01 cos^{2} (θ) - 0.12734 cos (θ) + 0.59461311

= - 1.01 cos^{2} (θ) - 0.12734 cos (θ) + 0.59461311

0.59461311 - 0.12734 cos (θ) - 1.01 cos^{2} (θ) = 0

Usando o método de substituição

0.59461311 - 0.12734 cos (θ) - 1.01 cos^{2} (θ) = 0

Sea:

cos (θ) = u

0.59461311 - 0.12734 u - 1.01 u^{2} = 0

0.59461311 - 0.12734 u - 1.01 u^{2} = 0 : u = - \frac{0.12734 + 2.41845244}{2.02}, u = \frac{2.41845244 - 0.12734}{2.02}

0.59461311 - 0.12734 u - 1.01 u^{2} = 0

Escrever na forma padrão

a x^{2} + b x + c = 0

- 1.01 u^{2} - 0.12734 u + 0.59461311 = 0

Resolver com a fórmula quadrática

- 1.01 u^{2} - 0.12734 u + 0.59461311 = 0

Fórmula geral para equações de segundo grau:

Para

a = - 1.01, b = - 0.12734, c = 0.59461311

u_{1, 2} = \frac{- ( - 0.12734 ) \pm ( - 0.12734 ) ^{2} - 4 ( - 1.01 ) \cdot 0.59461311}{2 ( - 1.01 )}

u_{1, 2} = \frac{- ( - 0.12734 ) \pm ( - 0.12734 ) ^{2} - 4 ( - 1.01 ) \cdot 0.59461311}{2 ( - 1.01 )}

(- 0.12734)^{2} - 4 (- 1.01) \cdot 0.59461311 = 2.41845244

(- 0.12734)^{2} - 4 (- 1.01) \cdot 0.59461311

Aplicar a regra

- (- a) = a

= (- 0.12734)^{2} + 4 \cdot 1.01 \cdot 0.59461311

Aplicar as propriedades dos expoentes:

(- a)^{n} = a^{n},

n

é par

(- 0.12734)^{2} = 0.1273 4^{2}

= 0.1273 4^{2} + 4 \cdot 0.59461311 \cdot 1.01

Multiplicar os números:

4 \cdot 1.01 \cdot 0.59461311 = 2.40223 \dots

= 0.1273 4^{2} + 2.40223 \dots

0.1273 4^{2} = 0.0162154756

= 0.0162154756 + 2.40223 \dots

Somar:

0.0162154756 + 2.40223 \dots = 2.41845244

= 2.41845244

u_{1, 2} = \frac{- ( - 0.12734 ) \pm 2.41845244}{2 ( - 1.01 )}

Separe as soluções

u_{1} = \frac{- ( - 0.12734 ) + 2.41845244}{2 ( - 1.01 )}, u_{2} = \frac{- ( - 0.12734 ) - 2.41845244}{2 ( - 1.01 )}

u = \frac{- ( - 0.12734 ) + 2.41845244}{2 ( - 1.01 )} : - \frac{0.12734 + 2.41845244}{2.02}

\frac{- ( - 0.12734 ) + 2.41845244}{2 ( - 1.01 )}

Remover os parênteses:

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

= \frac{0.12734 + 2.41845244}{- 2 \cdot 1.01}

Multiplicar os números:

2 \cdot 1.01 = 2.02

= \frac{0.12734 + 2.41845244}{- 2.02}

Aplicar as propriedades das frações:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{0.12734 + 2.41845244}{2.02}

u = \frac{- ( - 0.12734 ) - 2.41845244}{2 ( - 1.01 )} : \frac{2.41845244 - 0.12734}{2.02}

\frac{- ( - 0.12734 ) - 2.41845244}{2 ( - 1.01 )}

Remover os parênteses:

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

= \frac{0.12734 - 2.41845244}{- 2 \cdot 1.01}

Multiplicar os números:

2 \cdot 1.01 = 2.02

= \frac{0.12734 - 2.41845244}{- 2.02}

Aplicar as propriedades das frações:

\frac{- a}{- b} = \frac{a}{b}

0.12734 - 2.41845244 = - (2.41845244 - 0.12734)

= \frac{2.41845244 - 0.12734}{2.02}

As soluções para a equação de segundo grau são:

u = - \frac{0.12734 + 2.41845244}{2.02}, u = \frac{2.41845244 - 0.12734}{2.02}

Substituir na equação

u = cos (θ)

cos (θ) = - \frac{0.12734 + 2.41845244}{2.02}, cos (θ) = \frac{2.41845244 - 0.12734}{2.02}

cos (θ) = - \frac{0.12734 + 2.41845244}{2.02}, cos (θ) = \frac{2.41845244 - 0.12734}{2.02}

cos (θ) = - \frac{0.12734 + 2.41845244}{2.02} : θ = arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn, θ = - arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn

cos (θ) = - \frac{0.12734 + 2.41845244}{2.02}

Aplique as propriedades trigonométricas inversas

cos (θ) = - \frac{0.12734 + 2.41845244}{2.02}

Soluções gerais para

cos (θ) = - \frac{0.12734 + 2.41845244}{2.02}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

θ = arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn, θ = - arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn

θ = arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn, θ = - arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn

cos (θ) = \frac{2.41845244 - 0.12734}{2.02} : θ = arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn, θ = 2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn

cos (θ) = \frac{2.41845244 - 0.12734}{2.02}

Aplique as propriedades trigonométricas inversas

cos (θ) = \frac{2.41845244 - 0.12734}{2.02}

Soluções gerais para

cos (θ) = \frac{2.41845244 - 0.12734}{2.02}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

θ = arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn, θ = 2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn

θ = arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn, θ = 2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn

Combinar toda as soluções

θ = arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn, θ = - arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn, θ = arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn, θ = 2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn

Verificar as soluções inserindo-as na equação original

Verificar as soluções inserindo-as em

sin (θ) = 0.6367 + 0.1 cos (θ)

Eliminar aquelas que não estejam de acordo com a equação.

Verificar a solução

arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn :

Verdadeiro

arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn

Inserir

n = 1

arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 π 1

Para

sin (θ) = 0.6367 + 0.1 cos (θ)

inserir

θ = arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 π 1

sin (arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 π 1) = 0.6367 + 0.1 cos (arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 π 1)

Simplificar

0.55340 \dots = 0.55340 \dots

\Rightarrow Verdadeiro

Verificar a solução

- arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn :

Falso

- arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn

Inserir

n = 1

- arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 π 1

Para

sin (θ) = 0.6367 + 0.1 cos (θ)

inserir

θ = - arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 π 1

sin (- arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 π 1) = 0.6367 + 0.1 cos (- arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 π 1)

Simplificar

- 0.55340 \dots = 0.55340 \dots

\Rightarrow Falso

Verificar a solução

arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn :

Verdadeiro

arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn

Inserir

n = 1

arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 π 1

Para

sin (θ) = 0.6367 + 0.1 cos (θ)

inserir

θ = arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 π 1

sin (arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 π 1) = 0.6367 + 0.1 cos (arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 π 1)

Simplificar

0.70738 \dots = 0.70738 \dots

\Rightarrow Verdadeiro

Verificar a solução

2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn :

Falso

2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn

Inserir

n = 1

2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 π 1

Para

sin (θ) = 0.6367 + 0.1 cos (θ)

inserir

θ = 2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 π 1

sin (2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 π 1) = 0.6367 + 0.1 cos (2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 π 1)

Simplificar

- 0.70738 \dots = 0.70738 \dots

\Rightarrow Falso

θ = arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn, θ = arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn

Mostrar soluções na forma decimal

θ = 2.55514 \dots + 2 πn, θ = 0.78578 \dots + 2 πn

Gráfico

Exemplos populares

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cos(x)= 10/21

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