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Popular Trigonometria >

4sin^2(x/3)=1

Solução

4 sin^{2} (\frac{x}{3}) = 1

Solução

x = \frac{π}{2} + 6 πn, x = \frac{5 π}{2} + 6 πn, x = \frac{7 π}{2} + 6 πn, x = \frac{11 π}{2} + 6 πn

Graus

x = 9 0^{\circ} + 108 0^{\circ} n, x = 45 0^{\circ} + 108 0^{\circ} n, x = 63 0^{\circ} + 108 0^{\circ} n, x = 99 0^{\circ} + 108 0^{\circ} n

Passos da solução

4 sin^{2} (\frac{x}{3}) = 1

Usando o método de substituição

4 sin^{2} (\frac{x}{3}) = 1

Sea:

sin (\frac{x}{3}) = u

4 u^{2} = 1

4 u^{2} = 1 : u = \frac{1}{2}, u = - \frac{1}{2}

4 u^{2} = 1

Dividir ambos os lados por

4

4 u^{2} = 1

Dividir ambos os lados por

4

\frac{4 u ^{2}}{4} = \frac{1}{4}

Simplificar

u^{2} = \frac{1}{4}

u^{2} = \frac{1}{4}

Para

x^{2} = f (a)

as soluções são

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

u = \frac{1}{4}, u = - \frac{1}{4}

\frac{1}{4} = \frac{1}{2}

\frac{1}{4}

Aplicar a seguinte propriedade dos radicais:

assumindo que

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Fatorar o número:

4 = 2^{2}

= 2^{2}

Aplicar as propriedades dos radicais:

2^{2} = 2

= 2

= \frac{1}{2}

Aplicar a regra

1 = 1

= \frac{1}{2}

- \frac{1}{4} = - \frac{1}{2}

- \frac{1}{4}

Simplificar

\frac{1}{4} : \frac{1}{2}

\frac{1}{4}

Aplicar a seguinte propriedade dos radicais:

assumindo que

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Fatorar o número:

4 = 2^{2}

= 2^{2}

Aplicar as propriedades dos radicais:

2^{2} = 2

= 2

= \frac{1}{2}

= - \frac{1}{2}

Aplicar a regra

1 = 1

= - \frac{1}{2}

u = \frac{1}{2}, u = - \frac{1}{2}

Substituir na equação

u = sin (\frac{x}{3})

sin (\frac{x}{3}) = \frac{1}{2}, sin (\frac{x}{3}) = - \frac{1}{2}

sin (\frac{x}{3}) = \frac{1}{2}, sin (\frac{x}{3}) = - \frac{1}{2}

sin (\frac{x}{3}) = \frac{1}{2} : x = \frac{π}{2} + 6 πn, x = \frac{5 π}{2} + 6 πn

sin (\frac{x}{3}) = \frac{1}{2}

Soluções gerais para

sin (\frac{x}{3}) = \frac{1}{2}

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}

\frac{x}{3} = \frac{π}{6} + 2 πn, \frac{x}{3} = \frac{5 π}{6} + 2 πn

\frac{x}{3} = \frac{π}{6} + 2 πn, \frac{x}{3} = \frac{5 π}{6} + 2 πn

Resolver

\frac{x}{3} = \frac{π}{6} + 2 πn : x = \frac{π}{2} + 6 πn

\frac{x}{3} = \frac{π}{6} + 2 πn

Multiplicar ambos os lados por

3

\frac{x}{3} = \frac{π}{6} + 2 πn

Multiplicar ambos os lados por

3

\frac{3 x}{3} = 3 \cdot \frac{π}{6} + 3 \cdot 2 πn

Simplificar

\frac{3 x}{3} = 3 \cdot \frac{π}{6} + 3 \cdot 2 πn

Simplificar

\frac{3 x}{3} : x

\frac{3 x}{3}

Dividir:

\frac{3}{3} = 1

= x

Simplificar

3 \cdot \frac{π}{6} + 3 \cdot 2 πn : \frac{π}{2} + 6 πn

3 \cdot \frac{π}{6} + 3 \cdot 2 πn

3 \cdot \frac{π}{6} = \frac{π}{2}

3 \cdot \frac{π}{6}

Multiplicar frações:

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

= \frac{π 3}{6}

Eliminar o fator comum:

3

= \frac{π}{2}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiplicar os números:

3 \cdot 2 = 6

= 6 πn

= \frac{π}{2} + 6 πn

x = \frac{π}{2} + 6 πn

x = \frac{π}{2} + 6 πn

x = \frac{π}{2} + 6 πn

Resolver

\frac{x}{3} = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{2} + 6 πn

\frac{x}{3} = \frac{5 π}{6} + 2 πn

Multiplicar ambos os lados por

3

\frac{x}{3} = \frac{5 π}{6} + 2 πn

Multiplicar ambos os lados por

3

\frac{3 x}{3} = 3 \cdot \frac{5 π}{6} + 3 \cdot 2 πn

Simplificar

\frac{3 x}{3} = 3 \cdot \frac{5 π}{6} + 3 \cdot 2 πn

Simplificar

\frac{3 x}{3} : x

\frac{3 x}{3}

Dividir:

\frac{3}{3} = 1

= x

Simplificar

3 \cdot \frac{5 π}{6} + 3 \cdot 2 πn : \frac{5 π}{2} + 6 πn

3 \cdot \frac{5 π}{6} + 3 \cdot 2 πn

3 \cdot \frac{5 π}{6} = \frac{5 π}{2}

3 \cdot \frac{5 π}{6}

Multiplicar frações:

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

= \frac{5 π 3}{6}

Multiplicar os números:

5 \cdot 3 = 15

= \frac{15 π}{6}

Eliminar o fator comum:

3

= \frac{5 π}{2}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiplicar os números:

3 \cdot 2 = 6

= 6 πn

= \frac{5 π}{2} + 6 πn

x = \frac{5 π}{2} + 6 πn

x = \frac{5 π}{2} + 6 πn

x = \frac{5 π}{2} + 6 πn

x = \frac{π}{2} + 6 πn, x = \frac{5 π}{2} + 6 πn

sin (\frac{x}{3}) = - \frac{1}{2} : x = \frac{7 π}{2} + 6 πn, x = \frac{11 π}{2} + 6 πn

sin (\frac{x}{3}) = - \frac{1}{2}

Soluções gerais para

sin (\frac{x}{3}) = - \frac{1}{2}

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}

\frac{x}{3} = \frac{7 π}{6} + 2 πn, \frac{x}{3} = \frac{11 π}{6} + 2 πn

\frac{x}{3} = \frac{7 π}{6} + 2 πn, \frac{x}{3} = \frac{11 π}{6} + 2 πn

Resolver

\frac{x}{3} = \frac{7 π}{6} + 2 πn : x = \frac{7 π}{2} + 6 πn

\frac{x}{3} = \frac{7 π}{6} + 2 πn

Multiplicar ambos os lados por

3

\frac{x}{3} = \frac{7 π}{6} + 2 πn

Multiplicar ambos os lados por

3

\frac{3 x}{3} = 3 \cdot \frac{7 π}{6} + 3 \cdot 2 πn

Simplificar

\frac{3 x}{3} = 3 \cdot \frac{7 π}{6} + 3 \cdot 2 πn

Simplificar

\frac{3 x}{3} : x

\frac{3 x}{3}

Dividir:

\frac{3}{3} = 1

= x

Simplificar

3 \cdot \frac{7 π}{6} + 3 \cdot 2 πn : \frac{7 π}{2} + 6 πn

3 \cdot \frac{7 π}{6} + 3 \cdot 2 πn

3 \cdot \frac{7 π}{6} = \frac{7 π}{2}

3 \cdot \frac{7 π}{6}

Multiplicar frações:

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

= \frac{7 π 3}{6}

Multiplicar os números:

7 \cdot 3 = 21

= \frac{21 π}{6}

Eliminar o fator comum:

3

= \frac{7 π}{2}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiplicar os números:

3 \cdot 2 = 6

= 6 πn

= \frac{7 π}{2} + 6 πn

x = \frac{7 π}{2} + 6 πn

x = \frac{7 π}{2} + 6 πn

x = \frac{7 π}{2} + 6 πn

Resolver

\frac{x}{3} = \frac{11 π}{6} + 2 πn : x = \frac{11 π}{2} + 6 πn

\frac{x}{3} = \frac{11 π}{6} + 2 πn

Multiplicar ambos os lados por

3

\frac{x}{3} = \frac{11 π}{6} + 2 πn

Multiplicar ambos os lados por

3

\frac{3 x}{3} = 3 \cdot \frac{11 π}{6} + 3 \cdot 2 πn

Simplificar

\frac{3 x}{3} = 3 \cdot \frac{11 π}{6} + 3 \cdot 2 πn

Simplificar

\frac{3 x}{3} : x

\frac{3 x}{3}

Dividir:

\frac{3}{3} = 1

= x

Simplificar

3 \cdot \frac{11 π}{6} + 3 \cdot 2 πn : \frac{11 π}{2} + 6 πn

3 \cdot \frac{11 π}{6} + 3 \cdot 2 πn

3 \cdot \frac{11 π}{6} = \frac{11 π}{2}

3 \cdot \frac{11 π}{6}

Multiplicar frações:

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

= \frac{11 π 3}{6}

Multiplicar os números:

11 \cdot 3 = 33

= \frac{33 π}{6}

Eliminar o fator comum:

3

= \frac{11 π}{2}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiplicar os números:

3 \cdot 2 = 6

= 6 πn

= \frac{11 π}{2} + 6 πn

x = \frac{11 π}{2} + 6 πn

x = \frac{11 π}{2} + 6 πn

x = \frac{11 π}{2} + 6 πn

x = \frac{7 π}{2} + 6 πn, x = \frac{11 π}{2} + 6 πn

Combinar toda as soluções

x = \frac{π}{2} + 6 πn, x = \frac{5 π}{2} + 6 πn, x = \frac{7 π}{2} + 6 πn, x = \frac{11 π}{2} + 6 πn

Gráfico

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