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(sin(30)+csc(30))/(sin^2(30)+cos^2(60))

Solução

\frac{sin ( 3 0 ^{\circ} ) + csc ( 3 0 ^{\circ} )}{sin ^{2} ( 3 0 ^{\circ} ) + cos ^{2} ( 6 0 ^{\circ} )}

5

Passos da solução

\frac{sin ( 3 0 ^{\circ} ) + csc ( 3 0 ^{\circ} )}{sin ^{2} ( 3 0 ^{\circ} ) + cos ^{2} ( 6 0 ^{\circ} )}

Utilizar a seguinte identidade trivial:

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 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{1}{2}

Utilizar a seguinte identidade trivial:

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

csc (3 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

= 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}

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

Simplificar

\frac{\frac{1}{2} + 2}{( \frac{1}{2} ) ^{2} + ( \frac{1}{2} ) ^{2}} : 5

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

Somar elementos similares:

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

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

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

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

Aplicar as propriedades dos expoentes:

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

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

Aplicar a regra

1^{a} = 1

1^{2} = 1

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

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

Simplificar

\frac{1}{2} + 2

em uma fração:

\frac{5}{2}

\frac{1}{2} + 2

Converter para fração:

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

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

Já que os denominadores são iguais, combinar as frações:

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

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

1 + 2 \cdot 2 = 5

1 + 2 \cdot 2

Multiplicar os números:

2 \cdot 2 = 4

= 1 + 4

Somar:

1 + 4 = 5

= 5

= \frac{5}{2}

= \frac{\frac{5}{2}}{2 \cdot \frac{1}{2 ^{2}}}

Aplicar as propriedades das frações:

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

= \frac{5}{2 \cdot 2 \cdot \frac{1}{2 ^{2}}}

Multiplicar os números:

2 \cdot 2 = 4

= \frac{5}{4 \cdot \frac{1}{2 ^{2}}}

2^{2} = 4

= \frac{5}{4 \cdot \frac{1}{4}}

Multiplicar

4 \cdot \frac{1}{4} : 1

4 \cdot \frac{1}{4}

Multiplicar frações:

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

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

Eliminar o fator comum:

4

= 1

= \frac{5}{1}

Aplicar a regra

\frac{a}{1} = a

= 5

= 5

\frac{cos ( 1 2 ^{\circ} )}{1.17}

8 \cdot cos^{2} (3 0^{\circ})

14 \cdot sin (4 5^{\circ})

- arctan (\frac{20}{200})

arccos (- 2.31)