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Popular >

sin^3(x)-0.25sin(x)=0

Solução

sin^{3} (x) - 0.25 sin (x) = 0

Solução

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

Graus

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Passos da solução

sin^{3} (x) - 0.25 sin (x) = 0

Usando o método de substituição

sin^{3} (x) - 0.25 sin (x) = 0

Sea:

sin (x) = u

u^{3} - 0.25 u = 0

u^{3} - 0.25 u = 0 : u = 0, u = - 0.5, u = 0.5

u^{3} - 0.25 u = 0

Fatorar

u^{3} - 0.25 u : u (u + 0.25) (u - 0.25)

u^{3} - 0.25 u

Fatorar o termo comum

u : u (u^{2} - 0.25)

u^{3} - 0.25 u

Aplicar as propriedades dos expoentes:

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

u^{3} = u^{2} u

= u^{2} u - 0.25 u

Fatorar o termo comum

u

= u (u^{2} - 0.25)

= u (u^{2} - 0.25)

Fatorar

u^{2} - 0.25 : (u + 0.25) (u - 0.25)

u^{2} - 0.25

Aplicar as propriedades dos radicais:

a = (a)^{2}

0.25 = (0.25)^{2}

= u^{2} - (0.25)^{2}

Aplicar a regra da diferença de quadrados:

x^{2} - y^{2} = (x + y) (x - y)

u^{2} - (0.25)^{2} = (u + 0.25) (u - 0.25)

= (u + 0.25) (u - 0.25)

= u (u + 0.25) (u - 0.25)

u (u + 0.25) (u - 0.25) = 0

Usando o princípio do fator zero: Se

ab = 0

então

a = 0

b = 0

u = 0 or u + 0.25 = 0 or u - 0.25 = 0

Resolver

u + 0.25 = 0 : u = - 0.5

u + 0.25 = 0

Mova

0.25

para o lado direito

u + 0.25 = 0

Subtrair

0.25

de ambos os lados

u + 0.25 - 0.25 = 0 - 0.25

Simplificar

u = - 0.5

u = - 0.5

Resolver

u - 0.25 = 0 : u = 0.5

u - 0.25 = 0

Mova

0.25

para o lado direito

u - 0.25 = 0

Adicionar

0.25

a ambos os lados

u - 0.25 + 0.25 = 0 + 0.25

Simplificar

u = 0.5

u = 0.5

As soluções são

u = 0, u = - 0.5, u = 0.5

Substituir na equação

u = sin (x)

sin (x) = 0, sin (x) = - 0.5, sin (x) = 0.5

sin (x) = 0, sin (x) = - 0.5, sin (x) = 0.5

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

sin (x) = - 0.5 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

sin (x) = - 0.5

Soluções gerais para

sin (x) = - 0.5

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

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

sin (x) = 0.5 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = 0.5

Soluções gerais para

sin (x) = 0.5

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

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

Combinar toda as soluções

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

Gráfico

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