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Popular Trigonometria >

csc(3θ)=6sin(3θ),0<θ<2pi

Solução

csc (3 θ) = 6 sin (3 θ), 0 < θ < 2 π

Solução

θ = \frac{0.42053 \dots}{3}, θ = \frac{π - 0.42053 \dots}{3}, θ = \frac{0.42053 \dots + 2 π}{3}, θ = \frac{3 π - 0.42053 \dots}{3}, θ = \frac{0.42053 \dots + 4 π}{3}, θ = \frac{5 π - 0.42053 \dots}{3}, θ = \frac{π + 0.42053 \dots}{3}, θ = \frac{- 0.42053 \dots + 2 π}{3}, θ = \frac{3 π + 0.42053 \dots}{3}, θ = \frac{- 0.42053 \dots + 4 π}{3}, θ = \frac{5 π + 0.42053 \dots}{3}, θ = \frac{- 0.42053 \dots + 6 π}{3}

Graus

θ = 8.03161 \dots^{\circ}, θ = 51.96838 \dots^{\circ}, θ = 128.03161 \dots^{\circ}, θ = 171.96838 \dots^{\circ}, θ = 248.03161 \dots^{\circ}, θ = 291.96838 \dots^{\circ}, θ = 68.03161 \dots^{\circ}, θ = 111.96838 \dots^{\circ}, θ = 188.03161 \dots^{\circ}, θ = 231.96838 \dots^{\circ}, θ = 308.03161 \dots^{\circ}, θ = 351.96838 \dots^{\circ}

Passos da solução

csc (3 θ) = 6 sin (3 θ), 0 < θ < 2 π

Subtrair

6 sin (3 θ)

de ambos os lados

csc (3 θ) - 6 sin (3 θ) = 0

Reeecreva usando identidades trigonométricas

csc (3 θ) - 6 sin (3 θ)

Utilizar a Identidade Básica da Trigonometria:

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

= csc (3 θ) - 6 \cdot \frac{1}{csc ( 3 θ )}

6 \cdot \frac{1}{csc ( 3 θ )} = \frac{6}{csc ( 3 θ )}

6 \cdot \frac{1}{csc ( 3 θ )}

Multiplicar frações:

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

= \frac{1 \cdot 6}{csc ( 3 θ )}

Multiplicar os números:

1 \cdot 6 = 6

= \frac{6}{csc ( 3 θ )}

= csc (3 θ) - \frac{6}{csc ( 3 θ )}

csc (3 θ) - \frac{6}{csc ( 3 θ )} = 0

Usando o método de substituição

csc (3 θ) - \frac{6}{csc ( 3 θ )} = 0

Sea:

csc (3 θ) = u

u - \frac{6}{u} = 0

u - \frac{6}{u} = 0 : u = 6, u = - 6

u - \frac{6}{u} = 0

Multiplicar ambos os lados por

u

u - \frac{6}{u} = 0

Multiplicar ambos os lados por

u

uu - \frac{6}{u} u = 0 \cdot u

Simplificar

uu - \frac{6}{u} u = 0 \cdot u

Simplificar

uu : u^{2}

uu

Aplicar as propriedades dos expoentes:

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

uu = u^{1 + 1}

= u^{1 + 1}

Somar:

1 + 1 = 2

= u^{2}

Simplificar

- \frac{6}{u} u : - 6

- \frac{6}{u} u

Multiplicar frações:

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

= - \frac{6 u}{u}

Eliminar o fator comum:

u

= - 6

Simplificar

0 \cdot u : 0

0 \cdot u

Aplicar a regra

0 \cdot a = 0

= 0

u^{2} - 6 = 0

u^{2} - 6 = 0

u^{2} - 6 = 0

Resolver

u^{2} - 6 = 0 : u = 6, u = - 6

u^{2} - 6 = 0

Mova

6

para o lado direito

u^{2} - 6 = 0

Adicionar

6

a ambos os lados

u^{2} - 6 + 6 = 0 + 6

Simplificar

u^{2} = 6

u^{2} = 6

Para

x^{2} = f (a)

as soluções são

x = f (a), - f (a)

u = 6, u = - 6

u = 6, u = - 6

Verifique soluções

Encontrar os pontos não definidos (singularidades):

u = 0

Tomar o(s) denominador(es) de

u - \frac{6}{u}

e comparar com zero

u = 0

Os seguintes pontos são indefinidos

u = 0

Combinar os pontos indefinidos com as soluções:

u = 6, u = - 6

Substituir na equação

u = csc (3 θ)

csc (3 θ) = 6, csc (3 θ) = - 6

csc (3 θ) = 6, csc (3 θ) = - 6

csc (3 θ) = 6, 0 < θ < 2 π : θ = \frac{arccsc ( 6 )}{3}, θ = \frac{π - arccsc ( 6 )}{3}, θ = \frac{arccsc ( 6 ) + 2 π}{3}, θ = \frac{3 π - arccsc ( 6 )}{3}, θ = \frac{arccsc ( 6 ) + 4 π}{3}, θ = \frac{5 π - arccsc ( 6 )}{3}

csc (3 θ) = 6, 0 < θ < 2 π

Aplique as propriedades trigonométricas inversas

csc (3 θ) = 6

Soluções gerais para

csc (3 θ) = 6

csc (x) = a \Rightarrow x = arccsc (a) + 2 πn, x = π - arccsc (a) + 2 πn

3 θ = arccsc (6) + 2 πn, 3 θ = π - arccsc (6) + 2 πn

3 θ = arccsc (6) + 2 πn, 3 θ = π - arccsc (6) + 2 πn

Resolver

3 θ = arccsc (6) + 2 πn : θ = \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

3 θ = arccsc (6) + 2 πn

Dividir ambos os lados por

3

3 θ = arccsc (6) + 2 πn

Dividir ambos os lados por

3

\frac{3 θ}{3} = \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

Simplificar

θ = \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

θ = \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

Resolver

3 θ = π - arccsc (6) + 2 πn : θ = \frac{π}{3} - \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

3 θ = π - arccsc (6) + 2 πn

Dividir ambos os lados por

3

3 θ = π - arccsc (6) + 2 πn

Dividir ambos os lados por

3

\frac{3 θ}{3} = \frac{π}{3} - \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

Simplificar

θ = \frac{π}{3} - \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

θ = \frac{π}{3} - \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

θ = \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}, θ = \frac{π}{3} - \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

Soluções para o intervalo

0 < θ < 2 π

θ = \frac{arccsc ( 6 )}{3}, θ = \frac{π - arccsc ( 6 )}{3}, θ = \frac{arccsc ( 6 ) + 2 π}{3}, θ = \frac{3 π - arccsc ( 6 )}{3}, θ = \frac{arccsc ( 6 ) + 4 π}{3}, θ = \frac{5 π - arccsc ( 6 )}{3}

csc (3 θ) = - 6, 0 < θ < 2 π : θ = \frac{π + arccsc ( 6 )}{3}, θ = \frac{- arccsc ( 6 ) + 2 π}{3}, θ = \frac{3 π + arccsc ( 6 )}{3}, θ = \frac{- arccsc ( 6 ) + 4 π}{3}, θ = \frac{5 π + arccsc ( 6 )}{3}, θ = \frac{- arccsc ( 6 ) + 6 π}{3}

csc (3 θ) = - 6, 0 < θ < 2 π

Aplique as propriedades trigonométricas inversas

csc (3 θ) = - 6

Soluções gerais para

csc (3 θ) = - 6

csc (x) = - a \Rightarrow x = arccsc (- a) + 2 πn, x = π + arccsc (a) + 2 πn

3 θ = arccsc (- 6) + 2 πn, 3 θ = π + arccsc (6) + 2 πn

3 θ = arccsc (- 6) + 2 πn, 3 θ = π + arccsc (6) + 2 πn

Resolver

3 θ = arccsc (- 6) + 2 πn : θ = - \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

3 θ = arccsc (- 6) + 2 πn

Simplificar

arccsc (- 6) + 2 πn : - arccsc (6) + 2 πn

arccsc (- 6) + 2 πn

Utilizar a seguinte propriedade:

arccsc (- x) = - arccsc (x)

arccsc (- 6) = - arccsc (6)

= - arccsc (6) + 2 πn

3 θ = - arccsc (6) + 2 πn

Dividir ambos os lados por

3

3 θ = - arccsc (6) + 2 πn

Dividir ambos os lados por

3

\frac{3 θ}{3} = - \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

Simplificar

θ = - \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

θ = - \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

Resolver

3 θ = π + arccsc (6) + 2 πn : θ = \frac{π}{3} + \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

3 θ = π + arccsc (6) + 2 πn

Dividir ambos os lados por

3

3 θ = π + arccsc (6) + 2 πn

Dividir ambos os lados por

3

\frac{3 θ}{3} = \frac{π}{3} + \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

Simplificar

θ = \frac{π}{3} + \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

θ = \frac{π}{3} + \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

θ = - \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}, θ = \frac{π}{3} + \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

Soluções para o intervalo

0 < θ < 2 π

θ = \frac{π + arccsc ( 6 )}{3}, θ = \frac{- arccsc ( 6 ) + 2 π}{3}, θ = \frac{3 π + arccsc ( 6 )}{3}, θ = \frac{- arccsc ( 6 ) + 4 π}{3}, θ = \frac{5 π + arccsc ( 6 )}{3}, θ = \frac{- arccsc ( 6 ) + 6 π}{3}

Combinar toda as soluções

θ = \frac{arccsc ( 6 )}{3}, θ = \frac{π - arccsc ( 6 )}{3}, θ = \frac{arccsc ( 6 ) + 2 π}{3}, θ = \frac{3 π - arccsc ( 6 )}{3}, θ = \frac{arccsc ( 6 ) + 4 π}{3}, θ = \frac{5 π - arccsc ( 6 )}{3}, θ = \frac{π + arccsc ( 6 )}{3}, θ = \frac{- arccsc ( 6 ) + 2 π}{3}, θ = \frac{3 π + arccsc ( 6 )}{3}, θ = \frac{- arccsc ( 6 ) + 4 π}{3}, θ = \frac{5 π + arccsc ( 6 )}{3}, θ = \frac{- arccsc ( 6 ) + 6 π}{3}

Mostrar soluções na forma decimal

θ = \frac{0.42053 \dots}{3}, θ = \frac{π - 0.42053 \dots}{3}, θ = \frac{0.42053 \dots + 2 π}{3}, θ = \frac{3 π - 0.42053 \dots}{3}, θ = \frac{0.42053 \dots + 4 π}{3}, θ = \frac{5 π - 0.42053 \dots}{3}, θ = \frac{π + 0.42053 \dots}{3}, θ = \frac{- 0.42053 \dots + 2 π}{3}, θ = \frac{3 π + 0.42053 \dots}{3}, θ = \frac{- 0.42053 \dots + 4 π}{3}, θ = \frac{5 π + 0.42053 \dots}{3}, θ = \frac{- 0.42053 \dots + 6 π}{3}

Gráfico

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