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sin(60)cos(60)tan(60)cot(60)sec(60)csc(60)

Solução

sin (6 0^{\circ}) cos (6 0^{\circ}) tan (6 0^{\circ}) cot (6 0^{\circ}) sec (6 0^{\circ}) csc (6 0^{\circ})

1

Passos da solução

sin (6 0^{\circ}) cos (6 0^{\circ}) tan (6 0^{\circ}) cot (6 0^{\circ}) sec (6 0^{\circ}) csc (6 0^{\circ})

Utilizar a seguinte identidade trivial:

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 0^{\circ})

sin (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

Utilizar a seguinte identidade trivial:

cos (6 0^{\circ}) = \frac{1}{2}

cos (6 0^{\circ})

cos (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{1}{2}

Utilizar a seguinte identidade trivial:

tan (6 0^{\circ}) = 3

tan (6 0^{\circ})

tan (x)

tabela de periodicidade com ciclo de

18 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= 3

Utilizar a seguinte identidade trivial:

cot (6 0^{\circ}) = \frac{3}{3}

cot (6 0^{\circ})

cot (x)

tabela de periodicidade com ciclo de

18 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

= \frac{3}{3}

Utilizar a seguinte identidade trivial:

sec (6 0^{\circ}) = 2

sec (6 0^{\circ})

sec (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

= 2

Utilizar a seguinte identidade trivial:

csc (6 0^{\circ}) = \frac{2 3}{3}

csc (6 0^{\circ})

csc (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

= \frac{2 3}{3}

= \frac{3}{2} \cdot \frac{1}{2} 3 \frac{3}{3} \cdot 2 \cdot \frac{2 3}{3}

Simplificar

\frac{3}{2} \cdot \frac{1}{2} 3 \frac{3}{3} \cdot 2 \cdot \frac{2 3}{3} : 1

\frac{3}{2} \cdot \frac{1}{2} 3 \frac{3}{3} \cdot 2 \cdot \frac{2 3}{3}

Multiplicar frações:

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

= \frac{3 \cdot 1 \cdot 3 \cdot 2 3 3 \cdot 2}{2 \cdot 2 \cdot 3 \cdot 3}

Eliminar o fator comum:

2

= \frac{3 \cdot 1 \cdot 3 3 3}{3 \cdot 3}

3 \cdot 1 \cdot 333 = 3^{4 \cdot \frac{1}{2}}

3 \cdot 1 \cdot 333

Aplicar as propriedades dos expoentes:

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

3333 = 3^{\frac{1}{2}} \cdot 3^{\frac{1}{2}} \cdot 3^{\frac{1}{2}} \cdot 3^{\frac{1}{2}} = 3^{\frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2}}

= 1 \cdot 3^{\frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2}}

Somar elementos similares:

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

= 1 \cdot 3^{4 \cdot \frac{1}{2}}

Multiplicar:

1 \cdot 3^{4 \cdot \frac{1}{2}} = 3^{4 \cdot \frac{1}{2}}

= 3^{4 \cdot \frac{1}{2}}

= \frac{3 ^{4 \cdot \frac{1}{2}}}{3 \cdot 3}

Multiplicar os números:

3 \cdot 3 = 9

= \frac{3 ^{4 \cdot \frac{1}{2}}}{9}

Fatorar

9 : 3^{2}

Fatorar

9 = 3^{2}

= \frac{3 ^{4 \cdot \frac{1}{2}}}{3 ^{2}}

Eliminar o fator comum:

3^{4 \cdot \frac{1}{2}}

= 1

= 1

10 sin (\frac{π}{6} 5)

sin (\frac{5}{6}) π

arcsin (cos (3 0^{\circ}))

tan (arccsc (2))

cos (\frac{5 π}{4}) + i \cdot sin (\frac{5 π}{4})