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Popular Trigonometria >

cot^2(2x-pi/3)= 1/3

Solução

cot^{2} (2 x - \frac{π}{3}) = \frac{1}{3}

Solução

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

Graus

x = 6 0^{\circ} + 9 0^{\circ} n, x = 9 0^{\circ} + 9 0^{\circ} n

Passos da solução

cot^{2} (2 x - \frac{π}{3}) = \frac{1}{3}

Usando o método de substituição

cot^{2} (2 x - \frac{π}{3}) = \frac{1}{3}

Sea:

cot (2 x - \frac{π}{3}) = u

u^{2} = \frac{1}{3}

u^{2} = \frac{1}{3} : u = \frac{1}{3}, u = - \frac{1}{3}

u^{2} = \frac{1}{3}

Para

x^{2} = f (a)

as soluções são

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

u = \frac{1}{3}, u = - \frac{1}{3}

Substituir na equação

u = cot (2 x - \frac{π}{3})

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

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

cot (2 x - \frac{π}{3}) = \frac{1}{3} : x = \frac{π}{3} + \frac{πn}{2}

cot (2 x - \frac{π}{3}) = \frac{1}{3}

Aplique as propriedades trigonométricas inversas

cot (2 x - \frac{π}{3}) = \frac{1}{3}

Soluções gerais para

cot (2 x - \frac{π}{3}) = \frac{1}{3}

cot (x) = a \Rightarrow x = arccot (a) + πn

2 x - \frac{π}{3} = arccot (\frac{1}{3}) + πn

2 x - \frac{π}{3} = arccot (\frac{1}{3}) + πn

Resolver

2 x - \frac{π}{3} = arccot (\frac{1}{3}) + πn : x = \frac{π}{3} + \frac{πn}{2}

2 x - \frac{π}{3} = arccot (\frac{1}{3}) + πn

Simplificar

arccot (\frac{1}{3}) + πn : \frac{π}{3} + πn

arccot (\frac{1}{3}) + πn

Utilizar a seguinte identidade trivial:

arccot (\frac{1}{3}) = \frac{π}{3}

x - 3 - 1 - \frac{3}{3} 0 \frac{3}{3} 13 arccot (x) \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} arccot (x) 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ}

= \frac{π}{3} + πn

2 x - \frac{π}{3} = \frac{π}{3} + πn

Mova

\frac{π}{3}

para o lado direito

2 x - \frac{π}{3} = \frac{π}{3} + πn

Adicionar

\frac{π}{3}

a ambos os lados

2 x - \frac{π}{3} + \frac{π}{3} = \frac{π}{3} + πn + \frac{π}{3}

Simplificar

2 x - \frac{π}{3} + \frac{π}{3} = \frac{π}{3} + πn + \frac{π}{3}

Simplificar

2 x - \frac{π}{3} + \frac{π}{3} : 2 x

2 x - \frac{π}{3} + \frac{π}{3}

Somar elementos similares:

- \frac{π}{3} + \frac{π}{3} = 0

= 2 x

Simplificar

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

\frac{π}{3} + πn + \frac{π}{3}

Agrupar termos semelhantes

= \frac{π}{3} + \frac{π}{3} + πn

Combinar as frações usando o mínimo múltiplo comum:

\frac{2 π}{3}

Aplicar a regra

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

= \frac{π + π}{3}

Somar elementos similares:

π + π = 2 π

= \frac{2 π}{3}

= \frac{2 π}{3} + πn

2 x = \frac{2 π}{3} + πn

2 x = \frac{2 π}{3} + πn

2 x = \frac{2 π}{3} + πn

Dividir ambos os lados por

2

2 x = \frac{2 π}{3} + πn

Dividir ambos os lados por

2

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

Simplificar

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

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

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

\frac{\frac{2 π}{3}}{2} + \frac{πn}{2}

\frac{\frac{2 π}{3}}{2} = \frac{π}{3}

\frac{\frac{2 π}{3}}{2}

Aplicar as propriedades das frações:

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

= \frac{2 π}{3 \cdot 2}

Multiplicar os números:

3 \cdot 2 = 6

= \frac{2 π}{6}

Eliminar o fator comum:

2

= \frac{π}{3}

= \frac{π}{3} + \frac{πn}{2}

x = \frac{π}{3} + \frac{πn}{2}

x = \frac{π}{3} + \frac{πn}{2}

x = \frac{π}{3} + \frac{πn}{2}

x = \frac{π}{3} + \frac{πn}{2}

cot (2 x - \frac{π}{3}) = - \frac{1}{3} : x = \frac{πn}{2} + \frac{π}{2}

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

Aplique as propriedades trigonométricas inversas

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

Soluções gerais para

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

cot (x) = - a \Rightarrow x = arccot (- a) + πn

2 x - \frac{π}{3} = arccot (- \frac{1}{3}) + πn

2 x - \frac{π}{3} = arccot (- \frac{1}{3}) + πn

Resolver

2 x - \frac{π}{3} = arccot (- \frac{1}{3}) + πn : x = \frac{πn}{2} + \frac{π}{2}

2 x - \frac{π}{3} = arccot (- \frac{1}{3}) + πn

Simplificar

arccot (- \frac{1}{3}) + πn : \frac{2 π}{3} + πn

arccot (- \frac{1}{3}) + πn

Utilizar a seguinte identidade trivial:

arccot (- \frac{1}{3}) = \frac{2 π}{3}

x - 3 - 1 - \frac{3}{3} 0 \frac{3}{3} 13 arccot (x) \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} arccot (x) 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ}

= \frac{2 π}{3} + πn

2 x - \frac{π}{3} = \frac{2 π}{3} + πn

Mova

\frac{π}{3}

para o lado direito

2 x - \frac{π}{3} = \frac{2 π}{3} + πn

Adicionar

\frac{π}{3}

a ambos os lados

2 x - \frac{π}{3} + \frac{π}{3} = \frac{2 π}{3} + πn + \frac{π}{3}

Simplificar

2 x - \frac{π}{3} + \frac{π}{3} = \frac{2 π}{3} + πn + \frac{π}{3}

Simplificar

2 x - \frac{π}{3} + \frac{π}{3} : 2 x

2 x - \frac{π}{3} + \frac{π}{3}

Somar elementos similares:

- \frac{π}{3} + \frac{π}{3} = 0

= 2 x

Simplificar

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

\frac{2 π}{3} + πn + \frac{π}{3}

Agrupar termos semelhantes

= πn + \frac{π}{3} + \frac{2 π}{3}

Combinar as frações usando o mínimo múltiplo comum:

π

Aplicar a regra

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

= \frac{π + 2 π}{3}

Somar elementos similares:

π + 2 π = 3 π

= \frac{3 π}{3}

Dividir:

\frac{3}{3} = 1

= π

= πn + π

2 x = πn + π

2 x = πn + π

2 x = πn + π

Dividir ambos os lados por

2

2 x = πn + π

Dividir ambos os lados por

2

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

Simplificar

x = \frac{πn}{2} + \frac{π}{2}

x = \frac{πn}{2} + \frac{π}{2}

x = \frac{πn}{2} + \frac{π}{2}

Combinar toda as soluções

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

Gráfico

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