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Popular >

sin(x/2)= 1/2 sin(x)

Solução

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

Solução

x = 4 πn, x = 2 π + 4 πn

Graus

x = 0^{\circ} + 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n

Passos da solução

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

Subtrair

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

de ambos os lados

sin (\frac{x}{2}) - \frac{1}{2} sin (x) = 0

Simplificar

sin (\frac{x}{2}) - \frac{1}{2} sin (x) : \frac{2 sin ( \frac{x}{2} ) - sin ( x )}{2}

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

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

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

Multiplicar frações:

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

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

Multiplicar:

1 \cdot sin (x) = sin (x)

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

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

Converter para fração:

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

= \frac{sin ( \frac{x}{2} ) \cdot 2}{2} - \frac{sin ( x )}{2}

Já que os denominadores são iguais, combinar as frações:

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

= \frac{sin ( \frac{x}{2} ) \cdot 2 - sin ( x )}{2}

\frac{2 sin ( \frac{x}{2} ) - sin ( x )}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 sin (\frac{x}{2}) - sin (x) = 0

Sea:

u = \frac{x}{2}

2 sin (u) - sin (2 u) = 0

Reeecreva usando identidades trigonométricas

- sin (2 u) + 2 sin (u)

Utilizar a identidade trigonométrica do arco duplo:

sin (2 x) = 2 sin (x) cos (x)

= - 2 sin (u) cos (u) + 2 sin (u)

2 sin (u) - 2 cos (u) sin (u) = 0

Fatorar

2 sin (u) - 2 cos (u) sin (u) : - 2 sin (u) (cos (u) - 1)

2 sin (u) - 2 cos (u) sin (u)

Fatorar o termo comum

- 2 sin (u)

= - 2 sin (u) (- 1 + cos (u))

- 2 sin (u) (cos (u) - 1) = 0

Resolver cada parte separadamente

sin (u) = 0 or cos (u) - 1 = 0

sin (u) = 0 : u = 2 πn, u = π + 2 πn

sin (u) = 0

Soluções gerais para

sin (u) = 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}

u = 0 + 2 πn, u = π + 2 πn

u = 0 + 2 πn, u = π + 2 πn

Resolver

u = 0 + 2 πn : u = 2 πn

u = 0 + 2 πn

0 + 2 πn = 2 πn

u = 2 πn

u = 2 πn, u = π + 2 πn

cos (u) - 1 = 0 : u = 2 πn

cos (u) - 1 = 0

Mova

1

para o lado direito

cos (u) - 1 = 0

Adicionar

1

a ambos os lados

cos (u) - 1 + 1 = 0 + 1

Simplificar

cos (u) = 1

cos (u) = 1

Soluções gerais para

cos (u) = 1

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

u = 0 + 2 πn

u = 0 + 2 πn

Resolver

u = 0 + 2 πn : u = 2 πn

u = 0 + 2 πn

0 + 2 πn = 2 πn

u = 2 πn

u = 2 πn

Combinar toda as soluções

u = 2 πn, u = π + 2 πn

Substituir na equação

u = \frac{x}{2}

\frac{x}{2} = 2 πn : x = 4 πn

\frac{x}{2} = 2 πn

Multiplicar ambos os lados por

2

\frac{x}{2} = 2 πn

Multiplicar ambos os lados por

2

\frac{2 x}{2} = 2 \cdot 2 πn

Simplificar

x = 4 πn

x = 4 πn

\frac{x}{2} = π + 2 πn : x = 2 π + 4 πn

\frac{x}{2} = π + 2 πn

Multiplicar ambos os lados por

2

\frac{x}{2} = π + 2 πn

Multiplicar ambos os lados por

2

\frac{2 x}{2} = 2 π + 2 \cdot 2 πn

Simplificar

x = 2 π + 4 πn

x = 2 π + 4 πn

x = 4 πn, x = 2 π + 4 πn

Exemplos populares

tan(θ)=(-3)/3

tan (θ) = \frac{- 3}{3}

csc(2θ)= 1/2 (sec(θ))(cos(θ))

csc (2 θ) = \frac{1}{2} (sec (θ)) (cos (θ))

sec^2(x)+2sec(x)-8=0

sec^{2} (x) + 2 sec (x) - 8 = 0

3cos^2(x)+4sin(x)-4=0

3 cos^{2} (x) + 4 sin (x) - 4 = 0

sin(x)= 7/15

sin (x) = \frac{7}{15}