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Popular Trigonometria >

sin(135)+cos(135)+tan(135)

Solução

sin (13 5^{\circ}) + cos (13 5^{\circ}) + tan (13 5^{\circ})

Solução

- 1

Passos da solução

sin (13 5^{\circ}) + cos (13 5^{\circ}) + tan (13 5^{\circ})

Reeecreva usando identidades trigonométricas:

tan (13 5^{\circ}) = \frac{sin ( 13 5 ^{\circ} )}{cos ( 13 5 ^{\circ} )}

tan (13 5^{\circ})

Utilizar a Identidade Básica da Trigonometria:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{sin ( 13 5 ^{\circ} )}{cos ( 13 5 ^{\circ} )}

= sin (13 5^{\circ}) + cos (13 5^{\circ}) + \frac{sin ( 13 5 ^{\circ} )}{cos ( 13 5 ^{\circ} )}

Simplificar

= \frac{sin ( 13 5 ^{\circ} ) cos ( 13 5 ^{\circ} ) + cos ^{2} ( 13 5 ^{\circ} ) + sin ( 13 5 ^{\circ} )}{cos ( 13 5 ^{\circ} )}

Utilizar a seguinte identidade trivial:

sin (13 5^{\circ}) = \frac{2}{2}

sin (13 5^{\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{2}{2}

Utilizar a seguinte identidade trivial:

cos (13 5^{\circ}) = - \frac{2}{2}

cos (13 5^{\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{2}{2}

= \frac{\frac{2}{2} ( - \frac{2}{2} ) + ( - \frac{2}{2} ) ^{2} + \frac{2}{2}}{- \frac{2}{2}}

Simplificar

\frac{\frac{2}{2} ( - \frac{2}{2} ) + ( - \frac{2}{2} ) ^{2} + \frac{2}{2}}{- \frac{2}{2}} : - 1

\frac{\frac{2}{2} ( - \frac{2}{2} ) + ( - \frac{2}{2} ) ^{2} + \frac{2}{2}}{- \frac{2}{2}}

Remover os parênteses:

(- a) = - a

= \frac{- \frac{2}{2} \cdot \frac{2}{2} + ( - \frac{2}{2} ) ^{2} + \frac{2}{2}}{- \frac{2}{2}}

- \frac{2}{2} \cdot \frac{2}{2} + (- \frac{2}{2})^{2} + \frac{2}{2} = - \frac{2}{2} \cdot \frac{2}{2} + (\frac{2}{2})^{2} + \frac{2}{2}

- \frac{2}{2} \cdot \frac{2}{2} + (- \frac{2}{2})^{2} + \frac{2}{2}

Aplicar as propriedades dos expoentes:

(- a)^{n} = a^{n},

n

é par

(- \frac{2}{2})^{2} = (\frac{2}{2})^{2}

= - \frac{2}{2} \cdot \frac{2}{2} + (\frac{2}{2})^{2} + \frac{2}{2}

= \frac{- \frac{2}{2} \cdot \frac{2}{2} + ( \frac{2}{2} ) ^{2} + \frac{2}{2}}{- \frac{2}{2}}

Aplicar as propriedades das frações:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{- \frac{2}{2} \cdot \frac{2}{2} + ( \frac{2}{2} ) ^{2} + \frac{2}{2}}{\frac{2}{2}}

Aplicar as propriedades das frações:

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

\frac{- \frac{2}{2} \cdot \frac{2}{2} + ( \frac{2}{2} ) ^{2} + \frac{2}{2}}{\frac{2}{2}} = \frac{( - \frac{2}{2} \cdot \frac{2}{2} + ( \frac{2}{2} ) ^{2} + \frac{2}{2} ) \cdot 2}{2}

= - \frac{( - \frac{2}{2} \cdot \frac{2}{2} + ( \frac{2}{2} ) ^{2} + \frac{2}{2} ) \cdot 2}{2}

Aplicar as propriedades dos radicais:

n a = a^{\frac{1}{n}}

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

= \frac{2 ( ( \frac{2}{2} ) ^{2} + \frac{2}{2} - \frac{2}{2} \cdot \frac{2}{2} )}{2 ^{\frac{1}{2}}}

Aplicar as propriedades dos expoentes:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{1}}{2 ^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}}

= 2^{- \frac{1}{2} + 1} (\frac{2}{2})^{2} + \frac{2}{2} - \frac{2}{2} \cdot \frac{2}{2}

Subtrair:

1 - \frac{1}{2} = \frac{1}{2}

= 2^{\frac{1}{2}} (\frac{2}{2})^{2} + \frac{2}{2} - \frac{2}{2} \cdot \frac{2}{2}

Aplicar as propriedades dos radicais:

a^{\frac{1}{n}} = n a

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

= - 2 (\frac{2}{2})^{2} + \frac{2}{2} - \frac{2}{2} \cdot \frac{2}{2}

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

\frac{2}{2} \cdot \frac{2}{2}

Multiplicar frações:

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

= \frac{2 2}{2 \cdot 2}

22 = 2

22

Aplicar as propriedades dos radicais:

a a = a

22 = 2

= 2

= \frac{2}{2 \cdot 2}

Multiplicar os números:

2 \cdot 2 = 4

= \frac{2}{4}

Eliminar o fator comum:

2

= \frac{1}{2}

(\frac{2}{2})^{2} = \frac{1}{2}

(\frac{2}{2})^{2}

Aplicar as propriedades dos expoentes:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 2 ) ^{2}}{2 ^{2}}

(2)^{2} : 2

Aplicar as propriedades dos radicais:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

Aplicar as propriedades dos expoentes:

(a^{b})^{c} = a^{b c}

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

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

\frac{1}{2} \cdot 2

Multiplicar frações:

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

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

Eliminar o fator comum:

2

= 1

= 2

= \frac{2}{2 ^{2}}

Eliminar o fator comum:

2

= \frac{1}{2}

= - 2 (\frac{1}{2} - \frac{1}{2} + \frac{2}{2})

Combinar as frações usando o mínimo múltiplo comum:

\frac{2}{2}

Aplicar a regra

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

= \frac{- 1 + 1 + 2}{2}

- 1 + 1 = 0

= \frac{2}{2}

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

Remover os parênteses:

(a) = a

= - \frac{2}{2} 2

Multiplicar frações:

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

= - \frac{2 2}{2}

22 = 2

22

Aplicar as propriedades dos radicais:

a a = a

22 = 2

= 2

= - \frac{2}{2}

Aplicar a regra

\frac{a}{a} = 1

= - 1

= - 1

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