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Popular Trigonometria >

sec(β/4)csc(β/4)=2csc(β/4)

Solução

sec (\frac{β}{4}) csc (\frac{β}{4}) = 2 csc (\frac{β}{4})

Solução

β = \frac{4 π}{3} + 8 πn, β = \frac{20 π}{3} + 8 πn

Graus

β = 24 0^{\circ} + 144 0^{\circ} n, β = 120 0^{\circ} + 144 0^{\circ} n

Passos da solução

sec (\frac{β}{4}) csc (\frac{β}{4}) = 2 csc (\frac{β}{4})

Subtrair

2 csc (\frac{β}{4})

de ambos os lados

sec (\frac{β}{4}) csc (\frac{β}{4}) - 2 csc (\frac{β}{4}) = 0

Fatorar

sec (\frac{β}{4}) csc (\frac{β}{4}) - 2 csc (\frac{β}{4}) : csc (\frac{β}{4}) (sec (\frac{β}{4}) - 2)

sec (\frac{β}{4}) csc (\frac{β}{4}) - 2 csc (\frac{β}{4})

Fatorar o termo comum

csc (\frac{β}{4})

= csc (\frac{β}{4}) (sec (\frac{β}{4}) - 2)

csc (\frac{β}{4}) (sec (\frac{β}{4}) - 2) = 0

Resolver cada parte separadamente

csc (\frac{β}{4}) = 0 or sec (\frac{β}{4}) - 2 = 0

csc (\frac{β}{4}) = 0 :

Sem solução

csc (\frac{β}{4}) = 0

csc (x) \leq - 1 or csc (x) \geq 1

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

sec (\frac{β}{4}) - 2 = 0 : β = \frac{4 π}{3} + 8 πn, β = \frac{20 π}{3} + 8 πn

sec (\frac{β}{4}) - 2 = 0

Mova

2

para o lado direito

sec (\frac{β}{4}) - 2 = 0

Adicionar

2

a ambos os lados

sec (\frac{β}{4}) - 2 + 2 = 0 + 2

Simplificar

sec (\frac{β}{4}) = 2

sec (\frac{β}{4}) = 2

Soluções gerais para

sec (\frac{β}{4}) = 2

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

\frac{β}{4} = \frac{π}{3} + 2 πn, \frac{β}{4} = \frac{5 π}{3} + 2 πn

\frac{β}{4} = \frac{π}{3} + 2 πn, \frac{β}{4} = \frac{5 π}{3} + 2 πn

Resolver

\frac{β}{4} = \frac{π}{3} + 2 πn : β = \frac{4 π}{3} + 8 πn

\frac{β}{4} = \frac{π}{3} + 2 πn

Multiplicar ambos os lados por

4

\frac{β}{4} = \frac{π}{3} + 2 πn

Multiplicar ambos os lados por

4

\frac{4 β}{4} = 4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Simplificar

\frac{4 β}{4} = 4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Simplificar

\frac{4 β}{4} : β

\frac{4 β}{4}

Dividir:

\frac{4}{4} = 1

= β

Simplificar

4 \cdot \frac{π}{3} + 4 \cdot 2 πn : \frac{4 π}{3} + 8 πn

4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Multiplicar

4 \cdot \frac{π}{3} : \frac{4 π}{3}

4 \cdot \frac{π}{3}

Multiplicar frações:

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

= \frac{π 4}{3}

= \frac{4 π}{3} + 4 \cdot 2 πn

Multiplicar os números:

4 \cdot 2 = 8

= \frac{4 π}{3} + 8 πn

β = \frac{4 π}{3} + 8 πn

β = \frac{4 π}{3} + 8 πn

β = \frac{4 π}{3} + 8 πn

Resolver

\frac{β}{4} = \frac{5 π}{3} + 2 πn : β = \frac{20 π}{3} + 8 πn

\frac{β}{4} = \frac{5 π}{3} + 2 πn

Multiplicar ambos os lados por

4

\frac{β}{4} = \frac{5 π}{3} + 2 πn

Multiplicar ambos os lados por

4

\frac{4 β}{4} = 4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn

Simplificar

\frac{4 β}{4} = 4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn

Simplificar

\frac{4 β}{4} : β

\frac{4 β}{4}

Dividir:

\frac{4}{4} = 1

= β

Simplificar

4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn : \frac{20 π}{3} + 8 πn

4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn

4 \cdot \frac{5 π}{3} = \frac{20 π}{3}

4 \cdot \frac{5 π}{3}

Multiplicar frações:

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

= \frac{5 π 4}{3}

Multiplicar os números:

5 \cdot 4 = 20

= \frac{20 π}{3}

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiplicar os números:

4 \cdot 2 = 8

= 8 πn

= \frac{20 π}{3} + 8 πn

β = \frac{20 π}{3} + 8 πn

β = \frac{20 π}{3} + 8 πn

β = \frac{20 π}{3} + 8 πn

β = \frac{4 π}{3} + 8 πn, β = \frac{20 π}{3} + 8 πn

Combinar toda as soluções

β = \frac{4 π}{3} + 8 πn, β = \frac{20 π}{3} + 8 πn

Gráfico

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