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1/(4sqrt(6))arctan(sqrt(2/3)tan(2pi))

Solução

\frac{1}{4 6} arctan (\frac{2}{3} tan (2 π))

0

Passos da solução

\frac{1}{4 6} arctan (\frac{2}{3} tan (2 π))

\frac{1}{4 6} arctan (\frac{2}{3} tan (2 π)) = \frac{6 arctan ( \frac{2}{3} tan ( 2 π ) )}{24}

\frac{1}{4 6} arctan (\frac{2}{3} tan (2 π))

Multiplicar frações:

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

= \frac{1 \cdot arctan ( \frac{2}{3} tan ( 2 π ) )}{4 6}

Multiplicar:

1 \cdot arctan (\frac{2}{3} tan (2 π)) = arctan (\frac{2}{3} tan (2 π))

= \frac{arctan ( \frac{2}{3} tan ( 2 π ) )}{4 6}

Racionalizar

\frac{arctan ( \frac{2}{3} tan ( 2 π ) )}{4 6} : \frac{6 arctan ( \frac{2}{3} tan ( 2 π ) )}{24}

\frac{arctan ( \frac{2}{3} tan ( 2 π ) )}{4 6}

Multiplicar pelo conjugado

\frac{6}{6}

= \frac{arctan ( \frac{2}{3} tan ( 2 π ) ) 6}{4 6 6}

466 = 24

466

Aplicar as propriedades dos radicais:

a a = a

66 = 6

= 4 \cdot 6

Multiplicar os números:

4 \cdot 6 = 24

= 24

= \frac{6 arctan ( \frac{2}{3} tan ( 2 π ) )}{24}

= \frac{6 arctan ( \frac{2}{3} tan ( 2 π ) )}{24}

= \frac{6 arctan ( \frac{2}{3} tan ( 2 π ) )}{24}

tan (2 π) = tan (0)

tan (2 π)

Reescrever

2 π

como

π \cdot 2 + 0

= tan (π 2 + 0)

Utilizar a periodicidade de

tan

tan (x + π \cdot k) = tan (x)

tan (π \cdot 2 + 0) = tan (0)

= tan (0)

= \frac{6 arctan ( \frac{2}{3} tan ( 0 ) )}{24}

Utilizar a seguinte identidade trivial:

tan (0) = 0

tan (0)

tan (x)

tabela de periodicidade com ciclo de

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= \frac{6 arctan ( \frac{2}{3} \cdot 0 )}{24}

= \frac{6 arctan ( \frac{2}{3} \cdot 0 )}{24}

Simplificar

= \frac{6 arctan ( 0 )}{24}

Utilizar a seguinte identidade trivial:

arctan (0) = 0

arctan (0)

x 0 \frac{3}{3} 13 arctan (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} arctan (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ}

= 0

= \frac{6 \cdot 0}{24}

Simplificar

= 0

sin (\frac{π}{3}) + cos^{2} (\frac{5 π}{6})

cot (4 5^{\circ}) + cot (9 0^{\circ})

arctan (\frac{- 791.18}{287.22})

6 \cdot sin (2 5^{\circ})

2 sin (\frac{3 π}{4})